# Inhomogeneous chiral condensation under rotation in the holographic QCD

Yidian Chen<EMAIL_ADDRESS>School of Nuclear Science and Technology,
University of Chinese Academy of Sciences, Beijing 100049, China  Danning Li
<EMAIL_ADDRESS>Department of Physics and Siyuan Laboratory, Jinan
University, Guangzhou 510632, P.R. China  Mei Huang<EMAIL_ADDRESS>School of Nuclear Science and Technology, University of Chinese Academy of
Sciences, Beijing 100049, China

###### Abstract

We investigate inhomogeneous chiral condensation under rotation considering
finite size effects and boundary conditions in the holographic QCD model. The
rotational suppression effect determined by $\Omega r$ is confirmed in the
holographic model which is not influenced by the boundary conditions. For
chiral condensation at the center, it is found that under Neumann boundary
condition the finite size exhibits two opposite effects, i.e., catalysis at
high temperatures and inverse catalysis at low temperatures. In contrast,
under Dirichlet boundary condition, the effect of finite size on condensation
is inverse catalysis, and small size induces a phase transition from
inhomogeneous to homogeneous phase. The temperature-angular velocity phase
diagrams of QCD are obtained for different boundary conditions and sizes, and
it is found that the critical temperature decreases with angular velocity.

##  I Introduction

The rotating matter is widespread in nature and can exhibit interesting
properties and phenomena which are different from those of ordinary matter.
For Bose-Einstein condensates or ultracold atom systems, the inhomogeneous
lattice patterns of quantized vortices are formed in a rapidly rotating
superfluid [1, 2]. In astrophysics, a spinning millisecond pulsar produces
continuous gravitational waves, which can be detected by LIGO and Virgo [3, 4,
5]. In non-central heavy-ion collision (HIC) experiments, relativistic
rotating quark-gluon plasmas (QGP) can be created with about
$10^{4}-10^{5}\hbar$ angular momentum [6, 7, 8], and the well-known chiral
vortical effect [9, 10, 11] and chiral vortical wave [12] may occur in the
chiral fluids due to anomalous transport phenomena induced by rotation.

The QGP fireballs are produced by the HIC, which can be under extreme
environments such as high temperature and/or density. For the non-central
collision case, the strong magnetic field and the high vorticity can be
produced in the early stage of QGP, and the local angular velocity can reach
0.2 GeV with a relatively long lifetime according to the transport model [13,
14, 15] and hydrodynamic [16] simulations. It is known that magnetic fields
yield interesting effects on QCD phase diagram, e.g., magnetic catalysis [17,
18, 19] and inverse magnetic catalysis [20, 21, 22]. It is expected that
analogous to magnetic fields, the rotation will also have some impact on the
chiral condensation thus on the phase structure of QCD. As opposed to the
magnetic field, the rotation not only affects the fermion pairing but also
presses the matter toward the edge of the system. In analogy to the Tolman-
Ehrenfest effect in the non-inertial frame, the condensation under rotation is
considered inhomogeneous, for example, the formation of vortices is a general
feature in rotating trapped Bose-Einstein condensates [23, 24].

For inhomogeneous chiral condensation formed by rotation, two factors are
worth paying attention to: the spatial distribution of rotational speed and
the finite size effect from the speed-of-light limit. Under the local density
approximation, the angular velocity $\Omega$ of the system can be considered
uniform, and in this case Refs.[25, 26, 27] studied radial coordinate $r$
dependent chiral condensation. These literatures pointed out that the rotation
can be interpreted as an effective chemical potential, manifesting as a local
suppression effect. Except near the boundary, the variation of condensation
with radial direction is mild, which verifies the soundness of the
approximation. In addition, Ref.[28] identified another inhomogeneous effect,
namely the centrifugal-like effect, which appears only in the
nonuniform-$\Omega$ case and comes from the contribution of
$\partial_{r}\Omega$.

The temperature-rotation $T-\Omega$ phase diagram of chiral condensation was
explored as shown in Ref.[26]. Similar to the temperature-chemical potential
phase diagram of QCD, the $T-\Omega$ diagram contains a new critical end point
located at $T_{CEP}\simeq 0.020$GeV and $\Omega_{CEP}\simeq 0.644$GeV, which
connects the crossover at high temperature and the first-order phase
transition at low temperature. In Ref.[26], the effective model with four-
fermion interactions was used to describe the pairing phenomena and its
corresponding Hamiltonian includes the angular velocity $\Omega$ and angular
momentum $J$ coupling terms $\Omega\cdot J$. Not surprisingly, the acting of
the angular velocity $\Omega$ and the chemical potential $\mu$ are pretty
comparable, owing to the similarity of the terms $\Omega\cdot J$ and the
chemical potential-conserved charge coupling term $\mu\cdot N$.

The finite size effect under rotation must be taken into account, and the
results are sensitive to the choice of boundary conditions. The effect of
finite size on phase transitions without rotation has been extensively studied
[29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. As in Ref.[39], chiral
condensation exhibits either catalysis or inverse catalysis for different
boundary conditions and shows a quantized first-order phase transition in the
case of periodic boundary condition. The boundary effects for the spatial
profile of condensation under rotation have been discussed in Ref.[27], and it
was found that the condensation shows as a plateau away from the boundary and
oscillating near the boundary.

The non-perturbative characteristics of QCD in the infrared (IR) region give a
great challenge in solving hadron physics and QCD phase structure. Various
non-perturbative methods, such as lattice QCD [41, 42, 43, 44], Dyson-
Schwinger equations [45, 46], and functional renormalization group equations
[47, 48, 49], are applied to solve this problem. At the end of the 20th
century, the discovery of the anti-de Sitter/conformal field theory (AdS/CFT)
correspondence or gauge/gravity duality [50, 51, 52] provided a new insight
for handling non-perturbative problems. According to the holographic
principle, a $D$-dimensional field theory is dual to the $(D+1)$-dimensional
gravitational theory, and the extra dimensions can be understood as
renormalization group flows [53]. Starting from string theory, various top-
down models, such as the $D_{3}-D_{7}$ model [54], the Witten-Sakai-Sugimoto
model [55, 56], and the STU model [57, 58, 59] have been widely studied to
detect the properties of strongly coupled theories. Another method known as
bottom-up approach, such as the hard-wall model [60], the soft-wall model
[61], the Gubser model [62, 63, 64, 65], the improved holographic QCD model
[66, 67, 68], the refined model [69], the Dudal model [70], and the dynamical
holographic QCD model (DhQCD) [71, 72, 73, 74] have extensively studied hadron
spectra, thermodynamic and transport properties of QCD matter, and QCD phase
transitions.

There are usually three ways to introduce rotation effects in holography as
follows. The first is to study it in the rotating Kerr black hole background
as in Refs.[75, 76, 77, 78, 79]. The second is to do a coordinate
transformation to a static black hole as in Refs.[80, 81, 82, 83]. The
confinement/deconfinement phase transition of QCD under rotation is studied in
this way in Refs.[84, 85]. The third one is to introduce rotation in the gauge
field and solve it under the probe approximation, as in Refs.[86, 87, 88, 89,
90]. In this paper, the third scenario is chosen since the chiral phase
transition of QCD under rotation is investigated.

The paper is organized as follows. First, the five-dimensional holographic
model is introduced in Sec.II. Next, the profile of chiral condensation under
rotation is investigated in Sec.III. Again, the effect of finite size on
condensation is considered in Sec.IV. Then, the temperature-angular velocity
phase diagram is obtained in Sec.V. Finally, a short summary and discussion is
presented in Sec.VI.

##  II 5d setup

The soft wall AdS/QCD model [61] offers a good scenario to study the chiral
phase transition, and it is quite convenient to extend the system to study the
transition under rotation. This holographic model could successfully describe
the chiral first-order phase transition, second-order phase transition, and
crossover, e.g. as shown in Refs. [91, 92, 93], by mapping a complex matrix-
valued scalar field $X^{\alpha\beta}$ to the 4D operator
$\langle\bar{q^{\alpha}}q^{\alpha}\rangle$, with $\alpha,\beta$ the flavor
indexes. In the model, the phase transition is realized by adding nonlinear
terms in the potential and introducing a proper profile of the dilaton field
$\Phi$. Since the rotation destroys spatial homogeneity, in principle, the
metric and fields should depend on the four-dimensional spacetime coordinates
$x^{\mu}$. However, due to the complexity of the full numerical solution of
general relativity, one might take the probe approximation, i.e. neglecting
the back-reaction to the background metric and considering the rotational
effect on the flavor sector only. Furthermore, we would assume that the
rotating axis is fixed. Then, it is natural to take the cylindrical
coordinate.

Taking the above consideration, we take the AdS-Schwarzschild metric as
following

$ds^{2}=\frac{L^{2}}{z^{2}}[-f(z)dt^{2}+\frac{dz^{2}}{f(z)}+dr^{2}+r^{2}d\theta^{2}+dx_{3}^{2}],$
(1)

where $L$ is the AdS radius , and $f(z)=1-(\frac{z}{z_{h}})^{4}$ is the
blackening factor with the black hole horizon $z_{h}$. In the case of flat
boundary, $L$ will be cancelled in the equation of motion, so we set it to be
$1$. According to the holographic dictionary, the temperature of the 4D system
is mapped to the Hawking temperature of the background black hole, i.e.
$T=\frac{1}{\pi z_{h}}$. Here we take the notation in which $z$ is the fifth
dimension, $x_{3}$ is the direction of the angular velocity $\Omega$ and $r$
is the distance to the rotating axis. As a result of the rotation, the probe
action involves a new gauge field $A_{M}$, which is dual to the non-zero
current operator $\mathcal{O}^{\mu}=\langle\bar{q}\gamma^{\mu}q\rangle$
induced by the rotation, besides the complex scalar field $X$. Finally, the
probe action is given as

$S_{M}=-\int d^{5}x\sqrt{-g}e^{-\Phi(z)}\Bigg{\\{}\text{{\rm
Tr}}[(D^{M}X)^{\dagger}(D_{M}X)+V_{X}(|X|)]+\frac{1}{4}F_{MN}F^{MN}\Bigg{\\}},$
(2)

where $g$ is the determinant of metric, $\Phi(z)$ is the dilaton field, the
covariant derivative is defined as $D_{M}X=\partial_{M}X-iA_{M}X$, and the
field strength is written as $F_{MN}=\partial_{M}A_{N}-\partial_{N}A_{M}$. In
this work, we will focus on the scalar background and $U(1)$ current, so the
$\text{SU}(2)$ gauge field in the original soft wall model has been neglected.

According to Refs.[91, 92], the chiral phase transition can be described by
the following form of dilaton field

$\Phi(z)=-\mu_{1}z^{2}+(\mu_{1}+\mu_{0})z^{2}\tanh(\mu_{2}z^{2}).$ (3)

Following Refs.[71, 72, 91, 92],The values of the three parameters
$\mu_{0},\mu_{1},\mu_{2}$ would be taken as $\mu_{0}=(0.43{\rm
GeV})^{2},\mu_{1}=(0.83{\rm GeV})^{2},\mu_{2}=(0.176{\rm GeV})^{2}$ , by
fitting the Regge slopes of light mesons, the vacuum value of the condensate
and the transition temperature of the chiral phase transition.

The spontaneous breaking of the $SU(N_{f})_{L}\times SU(N_{f})_{R}$ symmetry
to the $SU(N_{f})_{V}$ subgroup is realized by the non-vanishing vacuum
expectation value $X_{0}=\frac{\chi}{\sqrt{2N_{f}}}I_{N_{f}\times N_{f}}$ of
the complex scalar field $X$, with number of flavors $N_{f}$ and identity
matrix $I$. In this work, we would focus on the case with $N_{f}=2$. According
to the AdS/CFT dictionary, the UV asymptotic behavior of the scalar field
$\chi$ has the form $m_{q}\zeta z+...+\frac{\sigma}{\zeta}z^{3}$ with quark
mass $m_{q}$, quark condensation $\sigma$ and the normalization constant
$\zeta=\frac{\sqrt{3}}{2\pi}$ [94]. In the chiral limit $m_{q}=0$, the
potential term $V(\chi)$ of the scalar field determines transition order of
the phase transition, which could be first- or second-order. Therefore, the
following potential term $V(\chi)$ is considered in the paper

$V(\chi)\equiv{\rm
Tr}[V_{X}(|X|)]=\frac{m_{5}^{2}}{2}\chi^{2}+\upsilon_{3}\chi^{3}+\upsilon_{4}\chi^{4},$
(4)

and the phase transition is implemented by selecting the appropriate
parameters. In the potential, the five-dimensional mass square is
$m_{5}^{2}=-3$, based on the holographic dictionary, while a non-zero value of
the cubic coupling $\upsilon_{3}$ in Eq.(4) yields the first-order phase
transition. Moreover, the quartic coupling $\upsilon_{4}$ in Eq.(4) determines
the value of the quark condensation at zero temperature. In the paper, as in
Refs.[91, 92], two cases are chosen as $(\upsilon_{3},\upsilon_{4})=(0,8)$ and
$(\upsilon_{3},\upsilon_{4})=(-3,8)$, where the first one generates a second-
order phase transition and the last one gives a first-order phase transition
in the chiral limit.

By taking proper values of the model parameters and solving the equation of
motion, one could obtain the chiral condensate as a function of temperature.
The Panel.(a) and Panel.(b) of Fig.1 show the results for
$(\upsilon_{3},\upsilon_{4})=(-3,8)$ and $(\upsilon_{3},\upsilon_{4})=(0,8)$,
respectively. In Fig.1(a), the purple dashed line indicates the metastable
solution and the red dashed line denotes the unstable solution. In Fig.1(b),
the difference between the black solid and dashed lines is the selection of
the quark masses, which are 0 and 7 MeV, respectively. It can be shown that
the critical temperature $T_{c}\simeq 173.4$ MeV for the first-order phase
transition, and $T_{c}\simeq 150$ MeV for the second-order phase transition
and crossover. For Fig.1, they are specifically described in Ref.[91, 92] and
will not be further clarified here.

( a )  ( b )

Figure 1:  The chiral condensation as a function of temperature $T$ with the
scalar potential in Eq.(4). In Panel.(a), the parameters are chosen as
$(m_{q},\upsilon_{3},\upsilon_{4})=(0,-3,8)$, where the blue solid line, red
dashed line and purple dashed line represent the thermodynamic stable,
metastable and unstable states of the first-order phase transition,
respectively. The results of the second-order phase transition and crossover
are shown in Panel.(b) with parameters of $(\upsilon_{3},\upsilon_{4})=(0,8)$,
where the black solid and black dashed lines represent the quark masses are 0
and 7 MeV, respectively.

Considering a rotating system in which the direction of angular velocity
$\Omega$ is fixed, both the scalar field $\chi$ and the gauge field $A_{M}$
depend on the fifth dimensional coordinate $z$ and radial coordinate $r$, yet
independent of the polar coordinate $\theta$ and height coordinate $x_{3}$,
i.e.

$\chi=\chi(z,r),\qquad\qquad A_{M}=A_{M}(z,r).$ (5)

In the $A_{z}=0$ gauge, the only nonzero component of the gauge field is
$A_{\theta}$, which is dual to the polar current operator
$\langle\bar{q}\gamma^{\theta}q\rangle$. To maintain the conservation of the
vector current $j^{\mu}=\bar{q}\gamma^{\mu}q$ of the system,
$\bar{q}\gamma^{r}q=0$ and $\bar{q}\gamma^{x_{3}}q=0$ are chosen, i.e., the
system is in the steady state and no radial and vertical flows exist. And the
non-zero polar direction component $A_{\theta}$ is an effective polarization
term $\overrightarrow{\Omega}\cdot\overrightarrow{J}$ in the dual field theory
with angular momentum $\overrightarrow{J}$. Therefore, the equations of motion
of the action Eq.(2) are

$\displaystyle\partial_{z}(f\partial_{z}\chi)-f(\frac{1}{z}+\Phi^{\prime})\partial_{z}\chi+\partial_{r}^{2}\chi+\frac{\partial_{r}\chi}{r}+(\frac{3}{z^{2}}+\frac{f^{\prime}}{z}-\frac{3f}{z^{2}}-\frac{f\Phi^{\prime}}{z}-\frac{3\upsilon_{3}\chi}{z}-4\upsilon_{4}\chi^{2}-\frac{A_{\theta}^{2}}{r^{2}})\chi$
$\displaystyle=0$ (6)
$\displaystyle\partial_{z}(f\partial_{z}A_{\theta})-f(\frac{1}{z}+\Phi^{\prime})\partial_{z}A_{\theta}+\partial_{r}^{2}A_{\theta}-\frac{\partial_{r}A_{\theta}}{r}-\chi^{2}A_{\theta}$
$\displaystyle=0$ (7)

where $\chi$ is replaced by $z\chi$.

As in Refs.[86, 87, 88, 90], the appropriate boundary conditions are chosen in
the following form to solve the equations of motion. At the AdS conformal
boundary, the boundary conditions of the $\chi$ and $A_{\theta}$ fields are

$\chi|_{z=0}=m_{q}\zeta,\qquad\qquad A_{\theta}|_{z=0}=\Omega(r)r^{2},$ (8)

where the angular velocity $\Omega(r)$ is not necessarily constant, but
spatially dependent. In general, $\Omega(r)$ might be solved in a full back-
reaction model when choosing proper initial conditions. However, since we are
working in probe limit due to the complexity of getting the full solution, we
will take certain phenomenological configurations of $\Omega(r)$ instead.

In this paper, three possible cases of angular velocity configurations
$\Omega(r)$ are investigated: (i) $\Omega=$constant; (ii)
$\Omega(r)=\omega_{0}({\rm exp}[1.5(r-r_{0})^{2}]+1)^{-1}$; (iii)
$\Omega(r)=\omega_{0}({\rm exp}[(r-r_{0})]+1)^{-1}$. The cases (ii) and (iii)
are from Ref.[28], where $\omega_{0}$ and $r_{0}$ are parameters. Among them,
case (i) considers the rigid rotation of the system with a constant angular
velocity; case (ii) is a vortex-like angular velocity distribution; case (iii)
is an angular velocity distribution concentrated at the center. For the sake
of causality, the polar direction is restricted to a finite size, i.e., the
radius $R$ of the system. Therefore, at the edges of the system, the boundary
conditions are taken as

$\partial_{r}\chi|_{r=R}=0\quad{\rm(Neumann)}\quad{\rm
or}\quad\chi|_{r=R}=0\quad{\rm(Dirichlet)},\qquad\qquad
A_{\theta}|_{r=R}=\Omega(R)R^{2},$ (9)

where $\chi$ is regarded as two possible boundary conditions, i.e. Neumann or
Dirichlet boundary condition. The Neumann boundary condition is chosen such
that the order parameter is independent of $r$ in the region away from the
angular velocity distribution. If the system is bounded in a box, the
Dirichlet boundary condition can be taken into account. At the center of the
polar direction, the following boundary conditions are considered

$\partial_{r}\chi|_{r=0}=0,\qquad\qquad\partial_{r}A_{\theta}|_{r=0}=0,$ (10)

which guarantee the smoothness of the field configuration at the center. With
the above equations of motion and boundary conditions, the profile of the
chiral condensation under rotation can be obtained.

By comparing the free energy of the inhomogeneous and homogeneous phases, the
steady state of the system can be determined. According to the holographic
principle, the partition functions of the gravitation and the boundary field
theory are equivalent $\mathcal{Z}_{QCD}=\mathcal{Z}_{gravity}$, so the free
energy of the system can be extracted from the on-shell action

$\displaystyle\mathcal{F}$ $\displaystyle=$
$\displaystyle\int_{0}^{R}dr\int_{0}^{z_{h}}dz\sqrt{-g}e^{-\Phi}(-\frac{1}{2}\upsilon_{3}\chi^{3}-\upsilon_{4}\chi^{4}+\frac{1}{4}F^{2})$
(11)
$\displaystyle-\int_{0}^{z_{h}}dz\frac{1}{2}(\sqrt{-g}e^{-\Phi}\chi\partial^{r}\chi)\Big{|}_{0}^{R}-\int_{0}^{R}dr\frac{1}{2}(\sqrt{-g}e^{-\Phi}\chi\partial^{z}\chi\Big{|}_{0}^{z_{h}}).$

##  III chiral condensation under rotation

In this section, the spatial profile of the chiral condensation under rotation
is discussed. As mentioned in the previous section, due to the rotation of the
system, nonzero polar orientation current
$\langle\bar{q}\gamma^{\text{$\theta$}}q\rangle$ is induced, which further
leads to spatially dependent chiral condensation. It is worth stating that
different spatial distribution of angular velocity $\Omega(r)$ will lead to
different configurations of chiral condensation. Therefore, the three cases
(i), (ii), and (iii) introduced before are considered, i.e., uniform angular
velocity, vortex-like distribution of rotational speed and centered
distribution. Of course, the selection of parameters for cases (ii)
$\Omega(r)=\omega_{0}({\rm exp}[1.5(r-r_{0})^{2}]+1)^{-1}$ and (iii)
$\Omega(r)=\omega_{0}({\rm exp}[(r-r_{0})]+1)^{-1}$ need to satisfy the
causality constraint. Again as a consequence of the causality requirement, the
system is restricted to the radius $R$. Hence, in this section, the radius is
fixed to the typical quark gluon plasma scale 4 fm, i.e., 20${\rm GeV}^{-1}$,
so as to avoid the finite size effect, which will be discussed in the next
section. Furthermore, two possible boundary conditions for the scalar field
$\chi$ as stated in Eq.(9) are illustrated in Sec.III.1 and Sec.III.2,
respectively.

###  III.1 Neumann boundary condition

In this subsection, chiral condensation as a function of radial coordinate $r$
is studied. Since it is insignificant whether the phase transition is first-
order, second-order or crossover on the qualitative results, the example is
illustrated in the form of the potential term Eq.(4) with the parameter
$(m_{q},\upsilon_{3},\upsilon_{4})=(0,-3,8)$ chosen.

For the temperature $T=170$ MeV and the uniform angular velocity $\Omega=0.01$
GeV, the configuration of the condensation $\sigma(r)$ in space is shown in
Fig.2. Among them, the left panel is a three-dimensional plot, while the right
two-dimensional plot with the horizontal coordinate $r$ from 0 to 4 fm. As can
be seen in the figure, the condensation is largest at the center and decreases
with increasing radial coordinate $r$, similar to a swelling. The condensation
has a maximum value of $0.0626\leavevmode\nobreak\ {\rm
GeV}^{3}\simeq(0.397\leavevmode\nobreak\ {\rm GeV})^{3}$ at the center, and it
gradually decreases to $0.0564\leavevmode\nobreak\ {\rm
GeV}^{3}\simeq(0.383\leavevmode\nobreak\ {\rm GeV})^{3}$ with increasing $r$.
It is worth noting that even at the center, the value of the condensation does
not reach the non-rotational value $0.0628\leavevmode\nobreak\ {\rm
GeV}^{3}\simeq(0.398\leavevmode\nobreak\ {\rm GeV})^{3}$. It arises from the
combined effect of rotation and finite size of the system. For the effect of
finite size, it will be explained in the next section. Furthermore, the
concrete shape of the condensation depends on the form of the angular velocity
function and the boundary conditions of the scalar field at the edges. The
Neumann boundary condition (NBC) is chosen here, while the Dirichlet boundary
condition (DBC) will be shown in the next subsection.

( a )  ( b )

Figure 2:  3D and 2D plots of chiral condensation as a function of radial $r$
at $T=170$MeV and $\Omega=0.01$ GeV with NBC and
$(m_{q},\upsilon_{3},\upsilon_{4})=(0,-3,8)$. In Fig.(b), the black line
indicates the value of condensation at the same temperature without rotation
and finite size.

The deformation of condensation by different distributions of angular velocity
is shown in Fig.3(a) with the case (ii) $\Omega(r)=0.18({\rm
exp}[1.5(r-3)^{2}]+1)^{-1}$ and case (iii) $\Omega(r)=0.01({\rm
exp}[(r-10)]+1)^{-1}$. The dashed and solid lines in the figure indicate the
variation of angular velocity and condensation with radial coordinate $r$,
respectively. From the red line, it can be seen that the angular velocity
takes a maximum value at radial $r=0.6$ fm, meanwhile the condensation obtains
a minimum value. At the center and the edge $R$, the angular velocity tends to
0 and the condensation goes to a stable value. The results suggest that for
the case (ii), the bigger the angular velocity, the more noticeable the
suppression of condensation, in agreement with Ref.[28]. This conclusion can
be seen from $\frac{A_{\theta}^{2}}{r^{2}}$ in Eq.(6). As the angular velocity
increases, the term $\frac{A_{\theta}^{2}}{r^{2}}$ becomes more crucial, which
is equivalent to increasing the five-dimensional mass $m_{5}^{2}$ and thus
reducing the chiral condensation. In Ref.[28], it pointed out that it is
$\Omega r$ instead of the angular velocity $\Omega$ that determines the
suppression effect, and this conclusion is also reproduced in the holographic
model. As shown in Fig.3(b), the condensation profile for case (ii) with
different parameters $r_{0}$ are exhibited. It can be seen that although the
distribution of the angular velocity is similar except for shifting the
center, the condensation decreases significantly as the radial $r$ becomes
larger. It can be further seen from Eq.(6) that the term
$\frac{A_{\theta}^{2}}{r^{2}}$ is converted to $\Omega^{2}r^{2}$ when
$A_{\theta}\simeq\Omega r^{2}$ is considered, then it is $\Omega r$ rather
than the angular velocity $\Omega$ that increases the five-dimensional mass.

For the green line, the shape of the condensation with radial coordinate $r$
is similar to that of the red line, both getting a minimum at a particular
position and reaching a stable value at the center and the edges. The
difference is that the angular velocity in case (iii) is approximately non-
zero constant near the center, so the suppression effect is similar to case
(i). However, near the edges, the green line converges to the red line. It
should be noted that this result of case (iii) is not consistent with
Ref.[28], where the enhancement of condensation by the first derivative of the
angular velocity is not exhibited in the holographic model.

( a )  ( b )

Figure 3:  The chiral condensation as a function of radial $r$ at $T=170$ MeV
with NBC and $(m_{q},\upsilon_{3},\upsilon_{4})=(0,-3,8)$, where the solid and
dashed lines denote the profile of the condensation and the distribution of
the angular velocity, respectively. In (a), the three cases of angular
velocity distribution are: (i) $\Omega=0.01$, (ii) $\Omega(r)=0.18({\rm
exp}[1.5(r-10)^{2}]+1)^{-1}$ and (iii) $\Omega(r)=0.01({\rm
exp}[(r-10)]+1)^{-1}$. Fig.(b) represents case (ii) $\Omega(r)=0.18({\rm
exp}[1.5(r-r_{0})^{2}]+1)^{-1}$ with $r_{0}=0.6$ fm, 1 fm, 1.6 fm and 2 fm.

In the following, returning to the case of uniform angular velocity, the role
of different temperatures and angular velocities on condensation is
investigated. The shape of condensation with radial $r$ is taken in Fig.4(a)
for fixing the angular velocity to 0.01 GeV and varying the temperature to
110, 130, 150, 170 and 190 MeV. It can be seen from the figure that the shape
of the condensation tends to be the same at different temperatures and that
the change in condensation is small as $r$ increases. The value at the center
increases from $0.0626\leavevmode\nobreak\ {\rm
GeV}^{3}\simeq(0.397\leavevmode\nobreak\ {\rm GeV})^{3}$ to
$0.094\leavevmode\nobreak\ {\rm GeV}^{3}\simeq(0.454\leavevmode\nobreak\ {\rm
GeV})^{3}$ as the temperature decreases from 170 MeV to 110 MeV. And for the
red line in the figure, i.e., when the temperature reaches 190 MeV, the system
enters into the homogeneous chiral symmetric phase. Consequently, it can be
expected that as the temperature increases, there exists a certain critical
temperature $T_{c}$, across which the system will enter the homogeneous chiral
restoration state and the condensation will no longer depend on space.
Further, at least for first- and second-order phase transitions, the phase
transition can be defined in this inhomogeneous phase, i.e., when the
condensation at the full space or center enters into chiral restoration, and
thus the $T-\Omega$ phase diagram of the chiral condensation can be drawn.

( a )  ( b )

Figure 4:  The chiral condensation as a function of radial coordinate $r$ with
NBC and $(m_{q},\upsilon_{3},\upsilon_{4})=(0,-3,8)$. Here, the solid and
dashed lines indicate the fixed angular velocity $\Omega=0.01$GeV and
temperature $T=170$MeV, respectively.

Here, fixing the temperature at 170 MeV and varying the angular velocity at
0.001, 0.002, 0.005, 0.01, and 0.05 GeV, the variation of the shape of the
condensation with the angular velocity is explored and the results are shown
in Fig.4(b). It can be obtained that the increasing angular velocity
suppresses the condensation, especially at the edges. As the angular velocity
increases from 0.001 GeV to 0.01 GeV, the change at the center is negligible
and the condensation decreases from $0.0626\leavevmode\nobreak\ {\rm
GeV}^{3}\simeq(0.397\leavevmode\nobreak\ {\rm GeV})^{3}$ to
$0.0564\leavevmode\nobreak\ {\rm GeV}^{3}\simeq(0.383\leavevmode\nobreak\ {\rm
GeV})^{3}$ at the edges. The greater angular velocity means that greater
deformation for condensation. Similar to the chemical potential, the angular
velocity plays the inverse catalysis effect. It is worth remarking here that
simply reducing the angular velocity does not allow the condensation value to
reach the no-rotation case, because the finite size effect also remains. And
for the red line in the figure, i.e., when the angular velocity reaches 0.05
GeV, the system enters the homogeneous chiral restoration phase and the value
of condensation in space is 0, which is analogous to the previous assertion.

###  III.2 Dirichlet boundary condition

In this subsection, chiral condensation as a function of radial $r$ is
discussed when the scalar field at the edge takes the DBC. Similar to the
previous subsection, The potential term Eq.(4) with the parameters
$(m_{q},\upsilon_{3},\upsilon_{4})=(0,-3,8)$ is selected as an example for
illustration.

For a temperature of 170 MeV and an angular velocity of 0.01 GeV, the 3D and
2D plots of the spatial profile of the condensation are shown in Fig.5. From
the figure, it can be observed that the value of condensation changes very
slowly at radial coordinate less than 2 fm, while at radial coordinate greater
than 2 fm, it decreases rapidly until 0. The shape of the condensation on the
space is similar to a drum, which is distinct from the NBC. At the center, the
value of condensation is about $0.0625\leavevmode\nobreak\ {\rm
GeV}^{3}\simeq(0.396\leavevmode\nobreak\ {\rm GeV})^{3}$, which is comparable
to the results of NBC. Again, this phenomenon arises from the finite size
effect of the system. Therefore, for angular velocity is not so large, the
value of condensation near the center does not depend significantly on the
boundary conditions at the edge. The plateau profile of condensation near the
center is consistent with Ref.[27], further demonstrating the validity of the
local density approximation, while holography does not find oscillating
behavior at the edges $R$.

( a )  ( b )

Figure 5:  3D and 2D plots of chiral condensation as a function of radius $r$
at $T=170$MeV and $\Omega=0.01$ GeV with DBC and
$(m_{q},\upsilon_{3},\upsilon_{4})=(0,-3,8)$. In Fig.(b), the black line
indicates the value of condensation at the same temperature without rotation
and finite size.

The deformation of the angular velocity function on the condensation when the
parameters of cases (ii) and (iii) are chosen as in the previous subsection is
shown in Fig.6. For the red line, the condensation yields a local minimum at
the maximum angular velocity, but it tends to 0 near the edge $R$. This is not
essentially different from the conclusion of the NBC, and the behavior at the
edges is driven by the boundary condition. As can be seen in Fig.6(b), the
suppressive effect of $\Omega r$ is also seen in DBC, consistent with NBC. For
the green line, its behavior near the center is similar to that of the blue
line, while near the edge is similar to that of the red line. It is worth
mentioning that the centrifugal-like effect of condensation does not appear in
the holographic model for either Neumann or Dirichlet boundary conditions,
inconsistent with Ref.[28]. The possible reason for this is that the
introduction of rotation in the holographic model comes from the gauge field,
which acts analogous to the chemical potential and therefore causes the system
to tend to chiral restoration, reducing the value of condensation.

( a )  ( b )

Figure 6:  The figure shows the chiral condensation as a function of radial
$r$ at $T=170$ MeV with DBC and $(m_{q},\upsilon_{3},\upsilon_{4})=(0,-3,8)$,
where the solid and dashed lines denote the profile of the condensation and
the distribution of the angular velocity, respectively. In Fig.(a), the three
cases of angular velocity distribution are: (i) $\Omega=0.01$, (ii)
$\Omega(r)=0.18({\rm exp}[1.5(r-10)^{2}]+1)^{-1}$ and (iii)
$\Omega(r)=0.01({\rm exp}[(r-10)]+1)^{-1}$. Fig.(b) represents case (ii)
$\Omega(r)=0.18({\rm exp}[1.5(r-r_{0})^{2}]+1)^{-1}$ with $r_{0}=0.6$ fm, 1
fm, 1.6 fm and 2 fm.

The condensation functions for different temperatures at fixed angular
velocity $\Omega=0.01$ GeV are displayed in Fig.7(a). As we can see in the
figure, the value at the center increases from $0.0625\leavevmode\nobreak\
{\rm GeV}^{3}\simeq(0.396\leavevmode\nobreak\ {\rm GeV})^{3}$ to
$0.093\leavevmode\nobreak\ {\rm GeV}^{3}\simeq(0.453\leavevmode\nobreak\ {\rm
GeV})^{3}$ as the temperature decreases from 170 MeV to 110 MeV. The special
feature is that the "surface" of the "drum" condensation shrinks as the
temperature increases. The plateau surface is about 1.5 fm for a temperature
of 170 MeV, while the surface expands to 3 fm for a temperature of 130 MeV.
Therefore, it can be expected that at 0 temperature, the condensation is
approximately "homogeneous" except near the edges. In turn, it can be
predicted that, similar to NBCs, DBCs also have phase transitions, and their
effects on the $T-\Omega$ phase diagram will be mentioned in Sec.V.

( a )  ( b )

Figure 7:  The chiral condensation as a function of radial coordinate $r$ with
DBC and $(m_{q},\upsilon_{3},\upsilon_{4})=(0,-3,8)$. Here, the solid and
dashed lines indicate the fixed angular velocity $\Omega=0.01$GeV and
temperature $T=170$MeV, respectively.

When the temperature is fixed at 170 MeV, condensation with different angular
velocities is shown in Fig.7(b). Similarly, the angular velocity does not
change much for condensation at the center, but the variation can reach
$0.01\leavevmode\nobreak\ {\rm GeV}^{3}$ near the edges. It is noteworthy that
at an angular velocity of 0.05 GeV, the system recovers to a state of chiral
symmetry, which is similar to the NBC.

##  IV finite size effect

In the previous section, the chiral condensation under rotation, including
various angular velocities and temperatures, was discussed. There the size of
the system was fixed to $R=4{\rm fm}\simeq 20{\rm GeV}^{-1}$, which is
comparable to the size of the fireball created in the current experiments. As
mentioned above, in a rotating system, the size effect might be important.
Also, the exact size of the fireball is hard to be extracted. Thus, in this
section, we will study the finite-size effect on condensation. It is common
knowledge that the thermal and dense quark matter produced in heavy ion
collisions is bounded in a finite volume at the nuclear scale. Consequently,
it is highly essential to recognize the effect of finite size on the QCD
matter and phase structure, which has been extensively studied during the past
decades[29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. As shown in the
research of Ref.[39], under suitable boundary conditions, the catalysis or
inverse catalysis of chiral symmetry breaking occurs as the size of the system
decreases. Of course, these studies did not consider the inhomogeneous chiral
condensation on the spatial coordinate $x$. Therefore, the effect of finite
size on the spatially dependent condensation, induced by the rotation of the
system, would be an interesting issue.

The profile of chiral condensation under rotation has been obtained in the
holographic model. The results show that the effect of rotation, similar to
the chemical potential, is inverse catalysis, regardless of NBC or DBC, while
the boundary conditions only affect the condensation near the edges.
Nevertheless, when finite size effects are included, different boundary
conditions will produce different results, which are also comparable to the
NJL model. In the NJL model, as in Ref.[39], periodic and antiperiodic
boundary conditions enhance symmetry breaking and restore chiral symmetry,
respectively, as the system size decreases. Therefore, in the following, NBC
and DBC will be chosen to evaluate the effect of finite size on the chiral
condensation and phase structure.

###  IV.1 Catalysis and inverse catalysis

In this subsection, the effect of finite size on the condensation is
investigated, considering NBC at the edges. Here, the five different radius of
the system are taken into account as 1-5 fm, i.e., 5-25 ${\rm GeV}^{-1}$, and
marked with the colors red, orange, green, blue and purple, respectively. As
in the previous section, the potential Eq.(4) with
$(m_{q},\upsilon_{3},\upsilon_{4})=(0,-3,8)$ is exhibited as an example.

At an angular velocity of 0.01 GeV and a temperature $T=170$ MeV close to the
phase transition temperature $T_{c}$, the condensation as a function of the
radial coordinate $r$ for various sizes $R$ is shown in Fig.8(a). It can be
seen from the figure that the variation of radius $R$ has little modification
on the condensation profile under NBC. As the size of the system decreases,
the condensation is enhanced both at the center and the edges. Therefore, it
can be concluded that the decrease of the system radius catalyses the chiral
symmetry breaking, similar to the periodic boundary condition in the NJL
model. The enlarged portion of Fig.8(a) shows the value of condensation at the
center as a function of radius $R$. From the Fig.8(a), it can be obtained that
the catalysis effect of finite size is pretty weak under NBC.

( a )  ( b )

Figure 8:  The figures show the profiles of condensation for different radii
with NBC, $(m_{q},\upsilon_{3},\upsilon_{4})=(0,-3,8)$ and $\Omega=0.01$.
Among them, Fig.(a) and Fig.(b) are at $T=170$ MeV and $T=50$ MeV,
respectively.

When the temperature of the system is apparently below the critical
temperature $T_{c}$, the catalysis effect of finite size will be converted to
inverse catalysis in case of NBC. At the angular velocity of 0.01 GeV and
temperature of 50 MeV, the condensation as a function of the radial coordinate
$r$ is plotted in Fig.8(b). As can be seen in Fig.8(b), the condensation
diminishes with the decrease of the system size except near the edges. The
behavior at the edges is from the role of boundary conditions. For the
enlarged portion of Fig.8(a) and (b), it is clear that at the center, the
condensation value of the smaller size is higher than that of the larger size
at high temperature, whereas the opposite is true at low temperature. The
condensation value at the center becomes smaller as the system size decreases,
which is the opposite of what happens at high temperature. Furthermore, no
matter the catalysis effect or inverse catalysis effect of the finite size,
they are quite weak under NBC and have little change on the condensation
value.

From the results in Fig.8, it is found that for the condensation at the
center, the finite size exhibits two totally opposite effects, i.e., catalysis
at high temperature and inverse catalysis at low temperature, in case of NBC.
It can be further deduced that the condensation value at the center may
strengthen or weaken as the system size changes due to the combination of the
two effects when the temperature is neither high nor low.

###  IV.2 Phase transition induced by finite size

In this subsection, the effect of finite size on chiral condensation is
studied with DBC at the edges. Instead of the choices in the previous
subsection, the five radii $R$ are selected here as 2.2, 2.4, 3, 4 and 5 fm.
The reason for such choices is that for the radius $R=1$ fm, the system does
not have inhomogeneous chiral condensation at the temperature of 170 MeV and
the angular velocity of 0.01 GeV with
$(m_{q},\upsilon_{3},\upsilon_{4})=(0,-3,8)$.

The chiral condensation as a function of the radial coordinate $r$ with DBC is
presented in Fig.9. Fig.9(a) shows the condensation of the five radii selected
and Fig.9(b) gives the value of condensation at the center as a function of
radius $R$. From the figure, it can be concluded that the condensation in all
spaces diminishes as the size of the system decreases and the value of it near
the center has a significant decrease when the radius of the system is less
than 3 fm. This suggests that, for DBC, the effect of finite size on
condensation is inverse catalysis. Thus, it can be conjectured that when the
system size is small enough, the condensation will not have inhomogeneous
solutions, and the results of numerical calculations prove it. When setting
the radius $R=1$ fm, the condensation of the full space is 0, i.e., the system
is restored to the chiral symmetric phase. So it can be further postulated
that as the size decreases, the system will suffer a phase transition from the
inhomogeneous phase (its presence at a point in space where the chiral
condensation is not zero) to the homogeneous phase (chiral symmetric phase).

( a )  ( b )

Figure 9:  Fig.(a) shows the results of chiral condensation with different
radii $R$, at $T=170$ MeV and $\Omega=0.01$ GeV with DBC and
$(m_{q},\upsilon_{3},\upsilon_{4})=(0,-3,8)$. Fig.(b) shows the value of
condensation at the center as a function of the radius $R$.

This postulation that the reduction of the system size induces a phase
transition can be verified by comparing the difference between the free energy
of the inhomogeneous and homogeneous phases. The free energy difference
between the two phases is shown in Fig.10(c), and it can be noted that the
difference is positive with $R_{c}\lesssim 2.8$ fm and negative with
$R_{c}\gtrsim 2.8$ fm. Therefore, the homogeneous phase is more stable when
the system size is less than the critical size $R_{c}\simeq 2.8$ fm, and
conversely, the inhomogeneous phase is preferred. Since the condensation is
maximum at the center, the value here is chosen to determine whether the
system is in the symmetric phase. The value of the condensation at the center
is shown as a function of the size $R$ in Fig.10(a), where the blue solid and
purple dashed lines indicate the stable and metastable phases, respectively.
When the parameters of the potential term Eq.(4) are chosen as
$(m_{q},\upsilon_{3},\upsilon_{4},)=(7{\rm MeV},0,8)$, i.e., the case of
crossover, the phase transition induced by the finite size turns into
crossover and the pseudo-critical size is around $R_{c}\simeq 1$ fm as in
Fig.10(b). Consequently, the order of the phase transition induced by the
finite size is determined by the form of the potential term Eq.(4). For the
case where the potential parameters are the first-order phase transition, the
finite size induces a first-order phase transition, and so forth. In summary,
the system size $R$ acts similarly to the inverse of temperature $1/T$,
inducing a phase transition from inhomogeneous to homogeneous phase.

( a )  ( b )
(c)

Figure 10:  Fig.(a) and (b) show the results of chiral condensation at the
center of the system as a function of radius $R$. Here, Fig.(a) shows the
results with $(m_{q},\upsilon_{3},\upsilon_{4})=(0,-3,8)$ at $T=170$ MeV and
$\Omega=0.01$ GeV, while Fig.(b) is the results with
$(m_{q},\upsilon_{3},\upsilon_{4})=(7{\rm MeV},0,8)$ at $T=115$ MeV and
$\Omega=0.01$. Fig.(c) shows the difference in free energy between the
inhomogeneous and homogeneous phases as a function of radius $R$ for
$(m_{q},\upsilon_{3},\upsilon_{4})=(0,-3,8)$.

##  V phase diagram with the angular velocity $\Omega$

In this section, the phase diagram of chiral transition on the temperature-
angular velocity plane will be discussed. It should be emphasized that the
condensation values are different in all points of the space due to the
rotation of the system. In order to study the $T-\Omega$ phase diagram,
following the previous section, the condensation value at the center
$\sigma_{r=0}$ is chosen as the order parameter of the system. In addition, it
is known from the above that the finite size has an impact on the phase
diagram, especially when the DBC is selected. Hence, the size of the system is
fixed to 2 fm or 4 fm, which are the typical sizes of the fire balls created
in the current heavy ion collisions, in order to compare the finite size
effect.

In Sec.III, it has been shown that with the parameters
$(m_{q},\upsilon_{3},\upsilon_{4})=(0,-3,8)$, when the temperature or angular
velocity increases to certain critical values, the inhomogeneous solutions
becomes disfavored and the chiral symmetry is restored everywhere. Thus, the
system undergoes the chiral phase transition at those critical values.
Further, the critical temperature $T_{c}$ and the critical angular velocity
$\Omega_{c}$ of the phase transition can be determined by the difference of
the free energy between the inhomogeneous and homogeneous phases. The free
energy difference and the condensation at the center as a function of
temperature $T$ or angular velocity $\Omega$ are shown in Fig.11. In
particular, Fig.11(a) shows the condensation at the center as a function of
temperature $T$ for $\Omega=0.01$ GeV and $R=4$ fm with NBC; Fig.11(b) shows
condensation at the center as a function of angular velocity for $T=170$ MeV
and $R=4$ fm with NBC. And the free energy difference as a function of
temperature and angular velocity is shown in Fig.11(c) and (d), respectively.
Here, the blue solid and purple dashed lines denote the stable and metastable
phases, respectively. As can be seen from the figure, the critical temperature
$T_{c}\simeq 172.5$ MeV for $\Omega=0.01$ GeV drops about 1 MeV compared to
that without rotation, which is not a big modification. As for the fixed
temperature, the condensation at the center varies little for small angular
velocities and the system returns to homogeneous phase when the critical
angular velocity $\Omega_{c}\simeq 0.016$GeV is reached.

( a )  ( b )
( c )  ( d )

Figure 11:  In Fig.(a) and (b), the chiral condensation at the center as a
function of temperature $T$ or angular velocity $\Omega$ are shown. Here, the
solid blue line indicates the stable state while the purple dashed line
denotes the metastable state. Fig.(c) and (d) shows the difference in free
energy between the inhomogeneous and homogeneous phases as a function of
temperature $T$ or angular velocity $\Omega$.

When the parameters of the model are set to
$(m_{q},\upsilon_{3},\upsilon_{4})=(7{\rm MeV},0,8)$, the phase transition of
the chiral condensation at the center is also converted to crossover and is
shown in Fig.12. In Fig.12(a) shows the condensation at the center as a
function of temperature at $\Omega=0.05$GeV and $R=4$ fm with NBC, and
Fig.12(b) shows the condensation at the center as a function of angular
velocity at $T=150$ MeV and $R=4$ fm with NBC. It is easy to see that
$\sigma_{r=0}$ decreases slowly at low $T$ and small $\Omega$, and it
decreases fast near the transition temperature, showing a typical behavior of
crossover in a narrow range of $T$ or $\Omega$. Usually, one can define the
pseudo-transition temperature where the slope reaches its maximum, i.e.
$\frac{d^{2}\sigma_{r=0}}{dT^{2}}$ or
$\frac{d^{2}\sigma_{r=0}}{d\Omega^{2}}=0$.

( a )  ( b )

Figure 12:  Fig.(a) and (b) represent the chiral condensation values at the
center as a function of temperature $T$ and angular velocity $\Omega$,
respectively, with NBC and $(m_{q},\upsilon_{3},\upsilon_{4})=(7{\rm
MeV},0,8)$.

With the above discussion, for each temperature one can get a critical
$\Omega$ and the phase diagram of the chiral phase transition in $T-\Omega$
space can be obtained. In heavy ion collisions, the rotation of hot dense
matter may have measurable consequences for chiral symmetry
restoration/breaking. Therefore, the study of the $T-\Omega$ phase diagram of
QCD matter can demonstrate the possible influences. In the case of $N_{f}=2$,
the chiral phase transition from the lattice QCD simulation is crossover, so
the model parameters are chosen to be
$(m_{q},\upsilon_{3},\upsilon_{4})=(7{\rm MeV},0,8)$ in which the phase
transition temperature is close to 150MeV and the vacuum value of $\sigma$ is
about ($0.320{\rm GeV})^{3}$. Furthermore, as previously discussed, the finite
size affects the phase diagram and phase structure, so the radii of the system
are chosen to be 4 fm and 2 fm for comparison. In both cases, the value of the
uniform angular velocity is limited to 0.05GeV and 0.1GeV, in order not to
violate the causality. The final results for the phase diagram are shown in
Fig.13, where the red line and the blue line correspond to the radii $R=2$ fm
and $R=4$ fm, respectively. In the holographic model, the boundary conditions
of the scalar field $\chi$ can be selected as NBC and DBC at the edges, and
the results are listed in Fig.13 with dashed line and dotted line,
respectively. As can be seen from the figure, the critical temperarure would
decrease with the angular velocity, and such a $T-\Omega$ phase diagram is
similar to the $T-\mu$ phase diagram, which is also consistent with the
previous conclusion that the angular velocity is analogous to the chemical
potential. If the system size decreases, the shifting of the phase boundary
depends on the choice of boundary conditions, for NBC the boundary line moves
outward, while for DBC the boundary line moves inward. This is consistent with
the effect of finite size on condensation for different boundary conditions,
i.e., near the critical temperature, NBC and DBC behave as catalysis and
inverse catalysis, respectively. For NBC, the phase boundary line is slightly
higher than the case of DBC. In summary, the $T-\Omega$ phase diagrams are
only slightly different even when considering the boundary conditions and
system size. If we consider the angular velocity of about 0.01 GeV produced by
heavy ion collision, the temperature of phase transition is about 128 MeV,
which shows a significant effect of the rotation.

Figure 13:  The $T-\Omega$ phase diagram at fixed radius $R$ is shown in the
figure. Here, the red and blue lines correspond to radii of $R=2$ fm and 4 fm,
respectively, and the dashed and dotted lines represent NBC and DBC,
respectively.

In Ref.[26], the first-order phase transition occurs at low temperature high
angular velocity with critical end point located at $T_{CEP}\simeq 0.02$GeV
and $\Omega_{CEP}\simeq 0.644$GeV. Since the phase structure depends on the
radius $R$, to produce an angular velocity of 0.65 GeV either limit the system
size to less than 1.5${\rm GeV}^{-1}$ or consider the angular velocity of the
vortex configuration. For the former, the radius of the system is so small
that the finite size has a large impact on the final result. For the latter,
the uncertainty of the phase diagram for different configurations cannot be
estimated. For these reasons, phase diagram for large angular velocity is not
included in this paper.

##  VI conclusion and discussion

In this paper, the configuration of chiral condensation under rotation
considering finite size and boundary conditions, and the temperature-angular
velocity phase diagram are investigated and discussed in the holographic QCD
model. In the holographic model, the rotation always exhibits a suppression
effect determined by $\Omega r$, which is consistent with Ref.[28], while the
centrifugal-like effect of Ref.[28] does not appear. In addition, the
influence of Neumann and Dirichlet boundary conditions on the profile of
condensation is mainly focused on the edges. For the finite-size effect, it
depends on the choice of boundary conditions. For NBC, the condensation at the
center is catalysis at high temperature and inverse catalysis at low
temperature, as the radius of the system decreases. For DBC, chiral
condensation is inverse catalysis and the phase transition is induced by
decreasing the size. $T-\Omega$ phase diagrams are obtained for different
boundary conditions and system sizes, and the differences between them are not
significant. At the angular velocity of 0.01 GeV, which is typical in heavy-
ion collisions, the critical temperature of the phase transition is about 128
MeV.

In the holographic model, the condensation under rotation exhibits the local
suppression effect, regardless of the boundary conditions or rotational
velocity distribution that are selected. Three cases of rotational velocity
distribution are discussed: (i) constant angular velocity, (ii) angular
velocity of the vortex structure and (iii) angular velocity concentrated at
the center. For case (i), the shape of the condensation as a function of the
radial coordinate $r$ is similar to a swelling or a drum, for case (ii) the
condensation has a minimum value at the center of the vortex, and for case
(iii) the condensation decreases slightly near the center. The condensation
does not display the centrifugal-like effect as shown in Ref.[28], although
case (iii) or small angular velocities are given as input. The possible reason
is that the rotation is introduced through the polar part of the gauge field,
which acts similarly to the chemical potential. The condensation with DBC has
a plateau profile near the center, which is consistent with Ref.[27] and
confirms the validity of the local density approximation. However, near the
edge $R$, the model does not show the oscillating behavior, which is
inconsistent with [27].

Different from the configuration of inhomogeneous condensation, the finite
size effect with uniform angular velocity depends on the choice of boundary
conditions, similar to that discussed in Ref.[39]. For NBC, at slightly below
the critical temperature, the decreasing size enhances the condensation value
in the full space, especially at the center, which behaves as the catalysis
effect. However, away from the critical temperature, the finite size effect
transforms into inverse catalysis, which is analogous to the behavior of
magnetic fields. For DBC, the finite size shows the inverse catalysis effect
at any temperature. And, as the size decreases, the system exhibits phase
transitions from the chiral broken phase to the chiral restored phase. The
order of the phase transition depends on the form of the potential term in the
5D action, if the parameters of the potential term are chosen as first-order
phase transition then the phase transition is first-order, and so forth.

The temperature-angular velocity phase diagram of chiral condensation has been
studied in the holographic model. Since the condensation is inhomogeneous, the
condensation at the center is taken as the order parameter of the system. In
consideration of the fact that both finite size and boundary conditions can
influence the profile of condensation, we selected four cases with radii of 2
fm or 4 fm and NBC or DBC, as a comparison. The results show that although the
phase lines of the four cases do not overlap, the differences between them are
negligible. The temperature-angular velocity phase diagram at large angular
velocity are not discussed because the size of the QGP is taken into account.
If the vortex structure similar to the Abrikosov lattice is achieved in the
holographic QCD model, the exploration of the phase structure at higher
rotational speed will be possible.

###### Acknowledgements.

This work is supported in part by the China Postdoctoral Science Foundation
under Grant No. 2021M703169, the National Natural Science Foundation of China
(NSFC) Grant Nos 11725523, 11735007, 11805084 and the Strategic Priority
Research Program of Chinese Academy of Sciences under Grant Nos XDB34030000
and XDPB15, the start-up funding from University of Chinese Academy of
Sciences(UCAS), the Fundamental Research Funds for the Central
Universities,and Guangdong Pearl River Talents Plan under Grant No.
2017GC010480.

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