aainstitutetext: School of Physical Sciences, University of Chinese Academy of
Sciences,
Beijing 100049, China bbinstitutetext: Institute of Theoretical Physics,
Chinese Academy of Sciences,
Beijing 100190, Chinaccinstitutetext: Department of Physics, Beijing Normal
University,
Beijing 100875, China

# Baby skyrmion in two-component holographic superfluids

Shunhui Yao a,b Yu Tian a Peng Yang c Hongbao Zhang
<EMAIL_ADDRESS>Corresponding author<EMAIL_ADDRESS>Corresponding author<EMAIL_ADDRESS>Corresponding author:
<EMAIL_ADDRESS>
###### Abstract

In the two-component Ginzburg-Landau theory of superfluidity, a pair of
fractional vortices form a composite type of topological defect, usually
referred to as a baby skyrmion. In this paper, we initiate the construction of
such a baby skyrmion in the holographic model of two-component superfluids. As
a result, two types of baby skyrmion configurations are found, where the
monopole-type of one is constructed directly by solving the static equations
of motion while the dipole-type of one is obtained by resorting to the time
evolution method. In addition, we find that the existence of these two types
of baby skyrmion depends on the inter-component coupling, reminiscent of the
situation in the baby skyrmion model.

##  1 Introduction

Topological defects, which occur ubiquitously in spontaneous symmetry
breaking, have been discussed in various contexts ranging from cosmic strings
to superconductivity, where Landau’s paradigm of order parameters is commonly
adopted to study the mechanism of phase transition. In the 2D Bose-Einstein
condensates described by the Ginzburg-Landau (GL) theory, quantized vortices
are known as elementary topological defects that appear commonly in both
superconductor and superfluid systems. In superconductivity theories, the
quantum vortex is called Abrikosov vortexabrikosov1957magnetic , which
carries, according to London’s Law, a quantized magnetic flux, while in the
context of superfluids, the counterpart corresponds to Onsager-Feynman
quantization of superfluid velocityfeynman1955chapter ; onsager1949statistical
. Both cases of quantization could be attributed to the single-valuedness of
the order parameters, and thus could be classified by the topological charge,
called winding number.

To make a straightforward generalization of the ordinary GL theory, one can
extend the one-order-parameter model to the multi-component cases, where
fractional vortices are instead considered as the fundamental type of
topological defects, featuring the violation of the ordinary flux-quantization
or Onsager-Feynman quantization in one-order-parameter GL
theoriesBabaev:2001hv ; babaev2007violation ; 2020PhRvA.101a3630R ;
2021PhRvA.103b3311R . In particular, in the two-component case, such
topological defects can be understood as baby skyrmions by constructing a
mapping from one $S^{2}$ (the 2D spatial plane with infinity included) to the
other $S^{2}$ comprised of the 3D unit vectors with its three components given
by the expectation values of Pauli matrices in the two-component
condensatesGaraud:2013vya . Accordingly, the resulting baby skyrmions can be
characterized by the topological charge associated with the second homotopy
group of the sphere, called skyrmion numberPiette:1994mh . On the other hand,
via the AdS/CFT correspondenceMaldacena:1997re ; Gubser:1998bc ; Witten:1998qj
, the Abelian-Higgs model of holographic superfluids and superconductors was
introduced in Hartnoll:2008vx ; Hartnoll:2008kx ; Herzog:2009he , with the
bulk hairy black hole dual to the the phase of spontaneously broken $U(1)$
symmetry on the boundaryGubser:2008px . Since then, such a holographic model
has been extensively studied. In particular, not only was the static
configuration of a single vortex constructed with its characteristic features
identifiedKeranen:2009re ; Montull:2009fe ; Domenech:2010nf ; Montull:2011im ;
Salvio:2012at ; Montull:2012fy ; Dias:2013bwa ; Salvio:2013jia ; Lan:2017 ,
but also the real time holographic vortex dynamics was exploredAdams:2012pj ;
Du:2014 ; Lan:2016 ; Wittmer:2020mnm ; Yan:2022jfc .

Just like the GL theory, the above holographic model of
superfluids/superconductors can also be extended to the two-component casesHe
; Du:2015zcb , where the vortex-like topological defects were revealed and the
real time dynamics of vortices were investigatedWu:2015sqk ; Yang:2019ibe . A
natural question along this line is whether there exist baby skyrmions in such
two-component holographic superfluids/superconductors. The purpose of this
paper is intended to address this issue by focusing on the topological charge
$\mathcal{Q}=1$ baby skyrmion. As a result, not only do we succeed in
constructing the dipole-type of baby skyrmion, but also the monopole-type of
baby skyrmion. Furthermore, we find that the existence of such baby skyrmions
depends on the bulk inter-component coupling parameter $\nu$, where the
monopole-type of baby skyrmion can only exist in the region with
$0<\nu<\nu_{c}$ and the dipole-type of baby skyrmion can only exist in the
region in which $\nu$ is negative, which is reminiscent of the result for the
baby skyrmion model, where the existence of the analogous baby skyrmions also
depends on the specific interactionLeask:2021hzm .

This paper is organized as follows. In the subsequent section, we shall
introduce the holographic model of two component superfluids we adopt in this
paper and elaborate on the two types of baby skyrmion under our consideration.
In Section 3, we will construct the monopole-type baby skyrmion by solving the
corresponding static equations of bulk fields. While in Section 4, we shall
resort to the time evolution method to search for the static configuration of
the dipole-type baby skyrmion via dynamic evolution. We conclude our paper
with some discussions.

##  2 Baby skyrmion configurations in Holographic superfluids

###  2.1 Holographic two-component superfluid model

We first introduce the holographic model of two-component superfluids. As
such, we consider an Abelian-Higgs model in (3+1)D asymptotic anti-de-Sitter
spacetime with $L$ the AdS radiusYang:2019ibe ; Wu:2015sqk , i.e.,

$S=\int_{M}\sqrt{-g}d^{4}x\left[\frac{1}{2\kappa^{2}}\left(R+\frac{6}{L^{2}}\right)-\frac{1}{e^{2}}\left(\frac{1}{4}F_{\mu\nu}F^{\mu\nu}+(D^{\mu}\Psi)^{\dagger}D_{\mu}\Psi+V(\Psi)\right)\right].$
(1)

Here $\Psi=(\Psi_{1},\Psi_{2})$ is a scalar doublet,
$F_{\mu\nu}=2\partial_{[\mu}A_{\nu]}$ is the field strength of the $U(1)$
gauge field $A$, and $D_{\mu}=\partial_{\mu}-iA_{\mu}$ is the $U(1)$ covariant
derivative operator with $e$ and $\kappa=\sqrt{8\pi G_{N}}$ the coupling
constants of gauge field and gravitational field respectively, where $G_{N}$
is the Newton constant111Here for simplicity, we have implicitly assumed that
the couplings of $\Psi_{1}$ and $\Psi_{2}$ with the $U(1)$ gauge field are the
same.. In addition, we choose the potential as
$V(\Psi)=m_{1}^{2}|\Psi_{1}|^{2}+m_{2}^{2}|\Psi_{2}|^{2}+\nu|\Psi_{1}|^{2}|\Psi_{2}|^{2}$
with $m_{1}$ and $m_{2}$ the mass parameters of the scalar fields and $\nu$
the coupling parameter between the scalar doublet222 Unlike its Ginzburg-
Landau counterpart in Garaud:2013vya , the bulk potential here does not
include the inter-band Josephson coupling term
$V_{\textbf{Josephson}}\propto\Psi_{1}^{*}\Psi_{2}+\Psi_{2}^{*}\Psi_{1}$, as
the presence of this term is equivalent to a shift of mass parameters of the
two scalar fields after a diagonalization of the quadratic part of Lagrangian.
.

The model will be treated under the probe limit. Namely, the back reaction of
the matter field onto the background geometry is neglected, which can be
achieved as the coupling strength of gravitational field is small enough
compared with that of gauge field, i.e. $\kappa\to 0$ or $e\to\infty$.
Accordingly, the matter sector can be living in a fixed background geometry.
Such a background metric for the model of holographic superfluids at finite
temperature is given by the Schwarzschild-AdS black-brane

$ds^{2}=\frac{L^{2}}{z^{2}}\left(-f(z)dt^{2}+\frac{1}{f(z)}dz^{2}+dx^{2}+dy^{2}\right),$
(2)

where $f(z)=1-(z/z_{h})^{3}$ with $z_{h}$ the black hole event horizon. The
Hawking temperature of the black-brane is given by $T=3/(4\pi z_{h})$, which
is identified with the temperature of the dual superfluid system by
holography. Without loss of generality, below we shall work with the unit in
which $L=1$ and set $z_{h}=1$333Note that our theory is scale invariant, so
only the dimensionless ratio quantities are meaningful. Our findings can be
represented in terms of the corresponding dimensionless quantities simply by
multiplying the appropriate power of the temperature ($T=\frac{3}{4\pi}$). For
example, $\mu\rightarrow\mu/T,r\rightarrow rT,t\rightarrow
tT,\langle\mathcal{O}_{i}\rangle\rightarrow\langle\mathcal{O}_{i}\rangle/T^{\Delta_{+i}}$.
But this reconstruction does not affect the main result we want to convey in
this paper, so we simply present our result in terms of the dimensional
quantities. .

With the asymptotic expansion of the bulk scalar fields near the AdS boundary
$z=0$ into two modes, namely the non-normalizable mode $\phi_{+}$ and the
normalizable mode $\phi_{-}$,

$\Psi_{i}=z^{\Delta_{-i}}\phi_{-i}+z^{\Delta_{+i}}\phi_{+i}+\cdots,$ (3)

one can read off the expectation value of the dual boundary scalar operator
$\langle\mathcal{O}_{i}\rangle=\phi_{+i}$, where $\Delta_{\pm
i}=3/2\pm\sqrt{9/4+m_{i}^{2}}$ with $\Delta_{+i}$ the scaling dimension of the
corresponding scalar operator $\mathcal{O}_{i}$. In what follows, we shall
focus on the case with $m_{i}^{2}=-2$ for simplicity. While $\phi_{-i}$ on the
boundary is regarded as a classical source in the generating functional of the
correlation function of the scalar operator $\mathcal{O}_{i}$. When we
consider the vacuum condensation of the system, the source $\phi_{-i}$ is
switched off. Similar to the ordinary GL theory, the field $\phi_{+i}$ serves
naturally as the order parameter of the holographic superfluid system.
Likewise, we can perform an asymptotic expansion for the gauge field,

$\displaystyle A_{t}$ $\displaystyle=\mu-\rho z+\mathcal{O}(z^{2})$ (4)
$\displaystyle A_{i}$ $\displaystyle=a_{i}+b_{i}z+\mathcal{O}(z^{2}),\quad
i=x,y,$ (5)

where we have chosen the axial gauge $A_{z}=0$. According to the holographic
dictionary, $\mu$ and $\rho$ are interpreted as the chemical potential and
charge density on the boundary, respectively. In addition, the spatial
component $a_{i}$ is related to the superfluid velocity, and $b_{i}$
corresponds to the charge current density on the boundaryHerzog:2009he . As
for the homogeneous and isotropic holographic superfluid without the inter-
component coupling, it has been shown in Yang:2019ibe  that when
$\mu>\mu_{c}=4.07$, the vacuum expectation value of $\mathcal{O}_{i}$ becomes
nonzero, indicating that the system undergoes a spontaneous breaking of the
global $U(1)$ symmetry to the superfluid state from the normal fluid state.
For our purpose, below we shall work exclusively with $\mu=5.0$, which gives
rise to the superfluid phase for the whole regime of the parameter space of
the coupling $\nu$ we are considering.

###  2.2 Baby skyrmion constructed out of vortices

#### From vortex to fractional vortex

As mentioned in the introduction section, there exists a new type of
topological defect in the 2D two-component GL theories, the fractional vortex,
which differs from the ordinary vortex in the sense that the norm of the
scalar doublet is everywhere none-zeroGaraud:2013vya . In particular, the
asymptotic behavior of the order parameter in the vicinity of the fractional
vortex core goes like

$\displaystyle\quad\psi\sim\binom{C\neq
0}{r^{n}e^{in\theta}}\quad\text{or}\quad\binom{r^{n}e^{in\theta}}{C\neq
0}\quad\text{when }r\to 0,$ (6)

which is different from that in the vicinity of the ordinary vortex core

$\displaystyle\psi\sim(re^{i\theta})^{n}\binom{C_{1}}{C_{2}},\quad|C_{1}|,|C_{2}|<\infty\quad\text{when
}r\to 0,$ (7)

where $(r,\theta)$ is the local polar coordinate with the vortex core located
at $r=0$, and $n$ is an integer number. Accordingly, unlike the ordinary
vortex, where the two components share the same winding number $n$ near the
vortex core, the fractional vortex carries the winding number $n$ for one
component but carries no winding number for the other one. This justifies why
it is named the fractional vortex.

#### From fractional vortex to baby skyrmion

According to Babaev:2008zd ; Babaev:2001zy ; Garaud:2013vya , the two-
component GL theory can be reformulated into a nonlinear sigma
modelFaddeev:1975tz ; faddeev1995einstein , where the baby skyrmion model of a
3D unit vector field $\mathbf{n}$ is referred to as the pseudo-spin:

$\mathbf{n}=\frac{\psi^{\dagger}\mathbf{\sigma}\psi}{\psi^{\dagger}\psi},\quad\sigma^{i}\leavevmode\nobreak\
\text{is the {i}th Pauli matrix}.$ (8)

A baby skyrmion is defined as a topologically nontrivial configuration of the
$\mathbf{n}$ field, which is characterized by the homotopy class of the
mapping $\varphi$ from the compactified coordinate space
$\mathbb{R}^{2}\cup{\infty}\simeq S^{2}$ to the target sphere $S^{2}$ :
$\mathbf{x}\mapsto\mathbf{n}$. The corresponding topological charge

$\mathcal{Q}:=\frac{1}{4\pi}\int_{\mathbb{R}^{2}}\varphi^{*}\omega=\frac{1}{4\pi}\int_{\mathbb{R}^{2}}\mathbf{n}\cdot\partial_{x}\mathbf{n}\times\partial_{y}\mathbf{n}\leavevmode\nobreak\
\mathbf{d}^{2}\mathbf{x},$ (9)

is called the skyrmion number , where $\omega$ is the standard volume form of
$S^{2}$. The skyrmion number can be an arbitrary integer, in accordance with
the fact that the second homotopy group of sphere is given by
$\pi_{2}(S^{2})\simeq\mathbb{Z}$.

As shown in Babaev:2008zd ; Babaev:2001zy ; Garaud:2013vya , a baby skyrmion
can be composed of two confined fractional vortices. Hence it is a composite
type of topological defect in the two-component GL theory. However, besides
the dipole-type baby skyrmion discussed in Garaud:2013vya , we argue that
there is also a monopole-type baby skyrmion. To be more specific, for the
$\mathcal{Q}=1$ dipole-type baby skyrmion, the order parameter has the profile

$\psi=\binom{f_{1}(\mathbf{r})e^{i\theta_{1}}}{f_{2}(\mathbf{r})e^{i\theta_{2}}},\quad\theta_{i}=\arctan{\frac{y-y_{i}}{x-x_{i}}},\quad
i=1,2,$ (10)

where $\mathbf{r}_{i}=(x_{i},y_{i})$ and $f_{i}$ stand for the center location
and the core function of the two fractional vortices such that
$f_{i}(\mathbf{r})\to|\mathbf{r}-\mathbf{r}_{i}|$ as
$\mathbf{r}\to\mathbf{r}_{i}$ while $f_{i}$ approaches the magnitude of the
homogeneous condensate at spatial infinity. In contrast, for the
$\mathcal{Q}=1$ monopole-type baby skyrmion, the order parameter has the
profile

$\psi=\binom{f_{1}(\mathbf{r})}{f_{2}(\mathbf{r})e^{i\theta_{2}}},\quad\theta_{2}=\arctan{\frac{y-y_{2}}{x-x_{2}}}$
(11)

with $f_{2}$ behaving as before but $f_{1}\to 0$ at the spatial infinity.

Both types of baby skyrmion carry a topological charge $\mathcal{Q}=1$. Here
we like to present a heuristic argument for it. According to Eq. (8), we have

$\mathbf{n}=(\frac{\zeta+\bar{\zeta}}{1+\zeta\bar{\zeta}},-i\frac{\zeta-\bar{\zeta}}{1+\zeta\bar{\zeta}},\frac{1-\zeta\bar{\zeta}}{1+\zeta\bar{\zeta}})$
(12)

with $\zeta$ the ratio of the two components of the scalar doublet, namely,
$\zeta=\psi_{2}/\psi_{1}$. Whence we can express $\zeta$ in terms of
$\mathbf{n}$ as follows

$\displaystyle\zeta$
$\displaystyle=\frac{\mathbf{n}^{1}+i\mathbf{n}^{2}}{1+\mathbf{n}^{3}},$ (13)

which amounts to saying that the mapping $\varphi:\mathbf{x}\mapsto\mathbf{n}$
can be represented by the complex function $z\to\zeta(z)$ from the coordinate
space to the target sphere $S^{2}\simeq\mathbb{CP}^{1}$ with $z=x+iy$ the
complex coordinate on $\mathbb{R}^{2}$. Then the skyrmion number $\mathcal{Q}$
can be written as

$\mathcal{Q}=\frac{1}{4\pi}\int\frac{2id\zeta
d\bar{\zeta}}{(1+\zeta\bar{\zeta})^{2}},$ (14)

which is identical to the degree of the mapping. For the case of monopole-type
baby skyrmion (11), the mapping is homotopic to the linear function
$\zeta=z-z_{c}$ with the vortex core $z_{c}$ and the infinity mapped to the
north pole and south pole of the target sphere, respectively. While for the
case of dipole-type baby skyrmion (10), the mapping is homotopic to the
function $\zeta=\frac{z-z_{1}}{z-z_{2}}$ with the two vortex cores $z_{1}$ and
$z_{2}$ mapped to the north pole and the south pole of the target sphere,
respectively. So both types of baby skyrmion have $1$ as the degree of the
mapping.

In the following sections, we are going to construct both types of baby
skyrmion in the holographic two-component superfluid model by numerics.

##  3 Monopole-type baby skyrmion

###  3.1 Ansatz for static solutions and relevant boundary conditions

Motivated by the previous discussions, the ansatz for the bulk static scalar
doublet is of the form:

$\Psi(z,r,\theta)=z\binom{u_{1}(z,r)}{u_{2}(z,r)e^{i\theta}},$ (15)

where the vortex is located at the origin. For the static gauge field sector,
the ansatz can also be made to respect the rotational symmetry as follows

$A_{t}=A_{t}(z,r),\leavevmode\nobreak\ A_{r}=0,\leavevmode\nobreak\
A_{\theta}=A_{\theta}(z,r).$ (16)

Substituting Eq. (15) and Eq. (16) into the equations of motion for the bulk
fields, one can obtain the following static equations

$\begin{split}\partial_{z}(f\partial_{z}u_{1})+\frac{1}{r}\partial_{r}(r\partial_{r}u_{1})&=u_{1}\left(-\frac{A_{t}^{2}}{f}+\frac{A_{\theta}^{2}}{r^{2}}+\nu\cdot
u_{2}^{2}+z\right),\\\
\partial_{z}(f\partial_{z}u_{2})+\frac{1}{r}\partial_{r}(r\partial_{r}u_{2})&=u_{2}\left(-\frac{A_{t}^{2}}{f}+\frac{(A_{\theta}-1)^{2}}{r^{2}}+\nu\cdot
u_{1}^{2}+z\right),\\\
f\partial_{z}^{2}A_{t}+\partial_{r}^{2}A_{t}+\frac{1}{r}\partial_{r}A_{t}&=2A_{t}(u_{1}^{2}+u_{2}^{2}),\\\
\partial_{z}\left(f\partial_{z}A_{\theta}\right)+\partial_{r}^{2}A_{\theta}-\frac{1}{r}\partial_{r}A_{\theta}&=2A_{\theta}(u_{1}^{2}+u_{2}^{2}).\end{split}$
(17)

To solve such equations, we are also required to impose some appropriate
boundary conditions. Speaking specifically, we shall impose the following
Dirichlet boundary conditions

$A_{t}\big{|}_{z=0}=\mu,\leavevmode\nobreak\
A_{\theta}\big{|}_{z=0}=0,\leavevmode\nobreak\ u_{1,2}\big{|}_{z=0}=0$ (18)

at the AdS boundary. In addition, to eliminate the singular terms in the above
equations of motion, we also impose

$A_{t}\big{|}_{z=1}=0$ (19)

on the black hole horizon. On the other hand, at the spatial boundaries, the
behaviors of $u_{1,2}$ are expected to be exchanged with each other, and the
regularity conditions of the fields near the coordinate singularity $r=0$ also
need to be incorporated. As a result, we have the corresponding boundary
conditions as follows

$\displaystyle u_{1}\to 0,\leavevmode\nobreak\ \partial_{r}u_{2}\to
0,\leavevmode\nobreak\ \partial_{r}A_{t}\to 0,\leavevmode\nobreak\
A_{\theta}\to 0\quad$ $\displaystyle\text{as}\quad r\to\infty,$ (20)
$\displaystyle u_{2}\to 0,\leavevmode\nobreak\ \partial_{r}u_{1}\to
0,\leavevmode\nobreak\ \partial_{r}A_{t}\to 0,\leavevmode\nobreak\
A_{\theta}\to 0\quad$ $\displaystyle\text{as}\quad r\to 0.$ (21)

###  3.2 Numerical results

Figure 1: The radial profile of the two components of the condensates for the
coupling $\nu=0.05$ at $\mu=5.0$.

The static solutions can be solved numerically by the pseudo-spectral method.
To this end, we make a truncation at $r=R$, where $R$ is chosen to be large
enough. For our purpose, we focus on $\mu=5.0$ and set $R=20$. It turns out
that the pseudo-spectral discretization with 25 Chebyshev modes in the $z$
direction and 48 Chebyshev modes in the $r$ direction is sufficient for us to
obtain the convergent static solutions.

Figure 2: The hedgehog view of the unit pseudo-spin vector field for
$\nu=0.03$ at $\mu=5.0$.

We first plot a typical radial profile of order parameters for the monopole-
type baby skymion in Fig. 1, where $\langle\mathcal{O}_{i}\rangle$ can be
obtained by computing $\partial_{z}u_{i}$ at $z=0$. As expected, the profile
of the second order parameter, which has a winding phase $e^{i\theta}$,
behaves like a vortex in the one-order-parameter superfluids, while the
condensate of the first order parameter is supported in the vicinity of the
vortex center like a Gaussian wave packet.

Our monopole-type baby skyrmion can also be visualized via a “hedgehog plot”
as usual. To make it, we first resort to the stereographic projection formula
(12) with $\zeta$ replaced by $z=x+iy$ to map the points on the $x$-$y$ plane
to the unit sphere in the 3D Euclidean space, and then attach each site with
the unit pseudo-spin vector given by Eq. (12). The resulting plot is
demonstrated in Fig. 2. As we see, the origin of the $x$-$y$ plane is mapped
to the north pole of the Riemann sphere with the pseudo-spin pointing north,
and the spatial infinity is mapped to the south pole with the pseudo-spin
pointing south, while other points on the plane are also mapped to the sphere
with the corresponding pseudo-spin pointing to various directions. Through
such a hedgehog visualization, one can intuitively see that the skyrmion
number of a monopole-type baby skyrmion is one, because the directions of
pseudo-spin vectors distributed on the Riemann sphere cover the target sphere
only once.444Numerical calculation of the integral (9) also confirms that the
skyrmion number $\mathcal{Q}$ of this monopole-type skyrmion is very close to
1 up to a numerical error of order $10^{-5}$.

Figure 3: The radial profile of the two components of the condensates with
$\nu$ varied form 0.15 to 0.23. at $\mu=5.0$ Figure 4: The maximum of the
first component of the condensates decreases to zero as $\nu$ increases to the
critical value $\nu_{c}=0.312$ at $\mu=5.0$, which signals the threshold for
the non-existence of the monopole-type baby skyrmion.

We have also investigated the effect of the inter-band coupling parameter
$\nu$ on the configuration of our monopole-type baby skyrmion. In Fig. 3, we
demonstrate the variation of the radial profile of the two condensates with
respect to $\nu$ separately. As one can see, with the increase of $\nu$, the
maximal value of $\langle\mathcal{O}_{1}\rangle$, located at the origin, is
decreased while the second condensate saturates the same maximal value in a
steeper manner. When we continue to crank up the coupling parameter $\nu$, the
first condensate will be further suppressed and then vanish at some critical
coupling $\nu_{c}\approx 0.312$, which is fixed by plotting the dependence of
the maximum of the first condensate on $\nu$ in Fig. 4. This means that the
monopole-type baby skyrmion no longer exists when $\nu\geq\nu_{c}$. By fitting
the curve, we see that near the critical point, there is a square root
behavior, that is, $\langle\mathcal{O}_{1}(r=0)\rangle\approx
7.23|\nu-\nu_{c}|^{1/2}$, typical of second order phase transitions.

##  4 Dipole-type baby skyrmion

In this section, we are going to study the $\mathcal{Q}=1$ dipole-type baby
skyrmion. The construction of the dipole-type baby skyrmion by directly
solving the static equation turns out to be rather complicated, for there is
no longer a rotational symmetry to exploit. As such, we shall instead resort
to the time evolution method to obtain a static configuration at the final
stage of the evolution. Actually, the presence of black hole horizon endows
the bulk system with a natural dissipation mechanism, which makes the time
evolution a natural dynamical relaxation process for obtaining a final static
state of interest.

To proceed, below we shall work in the ingoing Eddington-Finkelstein
coordinates, i.e.,

$ds^{2}=\frac{1}{z^{2}}\left(-f(z)dt^{2}-2dtdz+dr^{2}+r^{2}d\theta^{2}\right),$
(22)

where the full set of equations of motions for our matter sector are presented
in the appendix. In what follows, we shall elaborate on our evolution scheme
and the initial condition for our numerical simulations before we present the
relevant numerical results.

###  4.1 Evolution scheme and initial condition

In our evolution scheme, the evolution variables are chosen to be
$\\{\psi,A_{r},A_{\theta},\rho\\}$, where $\psi$ is given by $\Psi=z\psi$, and

$\rho=-\partial_{z}A_{t}\big{|}_{z=0}.$ (23)

Through a simple observation of the evolution equations in the appendix, one
can find that the evolution equations (31) to (35) can be cast into the common
form as

$\partial_{t}\mathcal{U}=\int_{0}^{z}\mathcal{E}(\mathcal{U},\partial\mathcal{U},A_{t})(z^{\prime})dz^{\prime}\quad\text{for}\quad\mathcal{U}\in\\{A_{\theta},A_{r},\psi\\},$
(24)

where the boundary conditions

$\partial_{t}\mathcal{U}\big{|}_{z=0}=0\quad\text{for}\quad\mathcal{U}\in\\{\psi,A_{\theta},A_{r}\\}$
(25)

have been used. On the other hand, the evolution equation for $\rho$ can be
obtained by restricting the mixed evolution equation (38) to the AdS boundary
$z=0$ as

$\partial_{t}\rho=-\left.\left(\frac{1}{r}\partial_{z}A_{r}+\partial_{z}\partial_{r}A_{r}+\frac{1}{r^{2}}\partial_{z}\partial_{\theta}A_{\theta}\right)\right|_{z=0}.$
(26)

$A_{t}$ is an intermediate variable, which can be determined by solving the
constraint equation (37) together with the following boundary conditions

$\displaystyle\partial_{z}A_{t}\big{|}_{z=0}=-\rho,\quad
A_{t}\big{|}_{z=0}=\mu.$ (27)

Thus the whole evolution system can be formulated as
$\partial_{t}\mathcal{U}=\mathcal{F}(\mathcal{U},A_{t}(\mathcal{U}))$, subject
to the boundary conditions at the truncation radius $r=R$ as follows,

$\partial_{r}\psi_{1,2}\big{|}_{r=R}=0,\quad A_{r,\theta}\big{|}_{r=R}=0,$
(28)

where $\mathcal{U}$ represents the full set of evolution variables.

To proceed, we still work with $\mu=5.0$, but take $R=15$ for our numerical
simulations. The reason why we take $R=15$ rather than $R=20$ as before for
the monopole-type baby skyrmion is that the accumulated numerical error, which
is monitored by the violation of the constraint equation (38) in the bulk,
will get uncontrolled in the polar coordinates as time goes on if the size of
the system is taken too large. With this set-up, we first discretize the
spatial directions by the pseudo-spectral method with $30$ Chebyshev modes in
the $z$ direction and $85$ Chebyshev modes in the $r$ direction as well as
$170$ Fourier modes in the $\theta$ direction, and then the evolution equation
can be integrated out by the $4$th order Runge-Kutta method with a typical
time step $\Delta t=0.002$, where the code is written in Juliajulia .

To set off the above evolution scheme, we need to prescribe an appropriate set
of initial conditions. For our purpose, we like to start with the initial
configuration homotopic to the dipole-type baby skyrmion. As such, the initial
value for $\psi$ is chosen as follows

$\psi(t=0,z,r,\theta)=\binom{\psi_{0}(z)f(r_{1}/r_{2})e^{i\theta_{1}}}{\psi_{0}(z)f(r_{2}/r_{1})e^{i\theta_{2}}},\quad\theta_{i}=\arctan{\frac{y-y_{i}}{x-x_{i}}},\quad
r_{i}=|\mathbf{r}-\mathbf{r}_{i}|,$ (29)

where $\psi_{0}(z)$ denotes the corresponding static configuration for the
homogeneous isotropic superfluid, and $f(\cdot)=\tanh(\frac{\cdot}{\xi})$ is a
core function of a single vortex with $\xi$ the typical size of a single
vortex. Similarly, the initial value for $\rho$ is also chosen as that for the
homogeneous isotropic superfluid. Finally, the initial data for $A_{r}$ and
$A_{\theta}$ are set to be zero for simplicity.

###  4.2 Numerical results

Our numerical simulations indicate that the final state will not settle down
to the dipole-type baby skyrmion when $\nu>0$.

Here we demonstrate the typical temporal evolution of our system in Fig. 5
with $\nu=-0.15$, where we plot the distribution of the condensate with the
color indicating the normalized value of the first condensate i.e.,
$\langle\mathcal{O}_{1}\rangle/\sqrt{\langle\mathcal{O}_{1}\rangle^{2}+\langle\mathcal{O}_{2}\rangle^{2}}$,
such that red color represents $1$ while the blue color stands for $0$. As one
can see, in each snapshot, there are a blue core and a red core. The center of
the blue core is exactly where the vortex of $\langle\mathcal{O}_{1}\rangle$
resides, which corresponds to the south pole of the target sphere. Similarly,
the center of the red core, which is the vortex center of
$\langle\mathcal{O}_{2}\rangle$, corresponds to the north pole.

Figure 5: Snapshots of the temporal evolution of the condensate for
$\nu=-0.15$ at $\mu=5.0$.

At the initial stage, the two poles of the dipole baby skyrmion are set very
close to each other, with $\mathbf{r}_{1}=-\mathbf{r}_{2}$ for the sake of
symmetry. At early times they are seen to begin repelling each other as well
as rotating anti-clockwise around each other. As time goes on, such a process
slows down gradually. After $t=500$, the system settles down to a final static
state, which can be regarded as the static baby skyrmion of the dipole type we
desire. This result also indicates the dynamical stability of the resulting
dipole-type baby skyrmion.

As depicted in Fig. 6, we also monitor the skyrmion number $\mathcal{Q}$ at
every time step of evolution. As one can see, the numerically calculated
topological charge $\mathcal{Q}$ is indeed very close to 1 with a negligible
error of order $10^{-4}$ at late times555The initial deviation from $1$ comes
essentially from the non-smoothness of the initial data, which dies out by the
black hole at late times..

Figure 6: Skyrmion number $\mathcal{Q}$ as a function of time. In late times,
$\mathcal{Q}$ converges to 1 as it should be the case.

It is noteworthy that we have never seen our temporal evolution for $\nu<0$
end up with the monopole-type baby skyrmion as the final state, which
indicates the non-existence of the monopole-type baby skyrmion in this
region666This also echos the fact we fail to construct the monopole type of
baby skyrmion in the $\nu<0$ region by solving the static equations of
motion..

##  5 Discussion

In this paper, we have succeeded in constructing both the monopole-type baby
skyrmion and the dipole-type one in the holographic two-component superfluid
model by directly solving the static equation of motion and resorting to the
temporal evolution scheme, respectively. In addition, our numerical result
indicates that the existence of such baby skyrmions depends on the interaction
between the two components. In particular, the monopole-type baby skyrmion can
exist only in the coupling parameter $0<\nu<\nu_{c}$ and the dipole-type baby
skyrmion can exist only in $\nu<0$. Such a result is reminiscent of that
obtained for the baby skyrmion model, where the existence of such baby
skyrmions also depends on the specific interaction termLeask:2021hzm . Hence,
it would be highly rewarding to develop an effective boundary description of
our bottom-up bulk model to deepen our understanding of such a similar
behavior.

We conclude our paper with a few additional issues worthy of further
investigation. The first direct generalizations of our work include examining
how our findings are affected by choosing different mass parameters $m_{1}$
and $m_{2}$ of our scalar doublet, giving up on the assumption that the gauge
couplings of $\Psi_{1}$ and $\Psi_{2}$ with the $U(1)$ gauge field are the
same, or taking into account the backreaction of the matter fields onto the
background geometry. Another direction is to extend our work to the three
spatial dimensions, where skyrmion corresponds to the local field
configuration which induces a topologically nontrivial mapping from
$\mathbb{R}^{3}$ to $S^{3}$skyrme1962unified ; skyrme1994selected ;
kotiuga2021continuum . There exist some attempts in the context of 3D two-
component GL theoryBattye:2001ec ; Kawakami:2012zw , but we expect to report
our result in the holographic model in the near future. Last but not least,
there have been a bunch of endeavors devoted to the statistical law of the
quantum turbulence of vortices such as Kolmogorov scaling lawAdams:2012pj ;
Lan:2016 , our current work opens a novel avenue for one to investigate the
statistical law of the quantum turbulence associated with these baby skyrmions
by holography, where some new features may be found.

###### Acknowledgements.

We are grateful to Yongliang Ma and Sven Bjarke Gudnason for helpful
discussions. This work is partially supported by the National Key Research and
Development Program of China Grant No. 2021YFC2203001 as well as the Natural
Science Foundation of China under Grants No. 11975235, No. 12035016, and No.
12075026.

##  Appendix A Appendix: Equations of motion in the ingoing Eddington-
Finkelstein coordinates

With the scalar doublet $\Psi=z(\psi_{1},\psi_{2})$ and the axial gauge
$A_{z}=0$, the whole set of equations of motion in the ingoing-Eddington-
Finkelstein coordinates can be classified as the following evolution equations

$\displaystyle\partial_{z}\partial_{t}\psi_{1}$
$\displaystyle=(-\Phi+\nu|\psi_{2}|^{2})\psi_{1}/2+(1/r^{2}-2iA_{r})\partial_{r}\psi_{1}/2+$
(30)
$\displaystyle\partial_{r}^{2}\psi_{1}/2-iA_{\theta}\partial_{\theta}\psi_{1}/r^{2}+\partial_{\theta}^{2}\psi_{1}/r^{2}+(iA_{t}+F_{r\theta})\partial_{z}\psi_{1}+F_{r\theta}\leavevmode\nobreak\
\partial_{z}^{2}\psi_{1}/2,$ (31)
$\displaystyle\partial_{z}\partial_{t}\psi_{2}$
$\displaystyle=(-\Phi+\nu|\psi_{1}|^{2})\psi_{2}/2+(1/r^{2}-2iA_{r})\partial_{r}\psi_{2}/2+$
(32)
$\displaystyle\partial_{r}^{2}\psi_{2}/2-iA_{\theta}\partial_{\theta}\psi_{2}/r^{2}+\partial_{\theta}^{2}\psi_{2}/r^{2}+(iA_{t}+F_{r\theta})\partial_{z}\psi_{2}+F_{r\theta}\leavevmode\nobreak\
\partial_{z}^{2}\psi_{2}/2,$ (33) $\displaystyle\partial_{z}\partial_{t}A_{r}$
$\displaystyle=-A_{r}(\psi^{\dagger}\psi)+\text{Im}(\psi^{\dagger}\partial_{r}\psi)-\partial_{\theta}F_{r\theta}/2r^{2}+\partial_{z}(f\partial_{z}A_{r}+\partial_{r}A_{t})/2,$
(34) $\displaystyle\partial_{z}\partial_{t}A_{\theta}$
$\displaystyle=-A_{\theta}(\psi^{\dagger}\psi)+\text{Im}(\psi^{\dagger}\partial_{\theta}\psi)-F_{r\theta}/2r^{2}+\partial_{r}F_{r\theta}/2+\partial_{z}(f\partial_{z}A_{\theta}+\partial_{\theta}A_{t})/2,$
(35)

where

$\Phi=A_{r}^{2}+A_{\theta}^{2}/r^{2}+z-i\partial_{z}A_{t}+i(A_{r}/r+\partial_{r}A_{r}+\partial_{\theta}A_{\theta}/r^{2}),\quad
F_{r\theta}=\partial_{r}A_{\theta}-\partial_{\theta}A_{r},$ (36)

as well as one constrain equation

$\partial_{z}^{2}A_{t}=-2\text{Im}(\psi^{\dagger}\partial_{z}\psi)+\partial_{z}(A_{r}/r+\partial_{r}A_{r}+\partial_{\theta}A_{\theta}/r^{2})$
(37)

and one mixed evolution equation

$\begin{gathered}\partial_{t}\partial_{z}A_{t}=-\frac{1}{r}\partial_{t}A_{r}-\partial_{t}\partial_{r}A_{r}-\frac{1}{r^{2}}\partial_{t}\partial_{\theta}A_{\theta}+\frac{f}{r}\partial_{z}A_{r}+f\partial_{z}\partial_{r}A_{r}+\frac{f}{r^{2}}\partial_{z}\partial_{\theta}A_{\theta}+\partial_{r}^{2}A_{t}\\\
+\frac{1}{r}\partial_{r}A_{t}+\frac{1}{r^{2}}\partial_{\theta}^{2}A_{t}+\text{Im}(\psi^{\dagger}\partial_{t}\psi)-2A_{t}\psi^{\dagger}\Psi-2f\text{Im}(\psi^{\dagger}\partial_{z}\psi).\end{gathered}$
(38)

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