aainstitutetext:  Department of Physics, Yamagata University, Kojirakawa-machi
1-4-12, Yamagata, Yamagata 990-8560, Japan bbinstitutetext:  Research and
Education Center for Natural Sciences, Keio University, 4-1-1 Hiyoshi,
Yokohama, Kanagawa 223-8521, Japan ccinstitutetext:  International Institute
for Sustainability with Knotted Chiral Meta Matter(SKCM2), Hiroshima
University, 1-3-2 Kagamiyama, Higashi-Hiroshima, Hiroshima 739-8511, Japan

YGHP-24-03

# Holographic Global Vortices with Novel Boundary Conditions

Markus A.G. Amano<EMAIL_ADDRESS>a,b,c Minoru Eto
<EMAIL_ADDRESS>
(March 28, 2024)

###### Abstract

The AdS/CFT correspondence has significantly impacted the study of strongly
coupled systems, providing insights into various condensed matter phenomena
through its holographic duality. This paper introduces an alternative approach
to the breaking of the global $U(1)$ symmetry in the bulk in two
asymptotically AdS spacetimes: AdS plus hard wall and AdS Blackbrane. We
explore a $(3+1)$-dimensional bulk $U(1)$ symmetry-breaking phase vacuum
within a global $U(1)$ $\phi^{4}$ field theory, without the gauge field
present in prior models. We find the symmetry-breaking vacuum requires that
the mass squared is proportional to the quartic coupling. We also investigate
numerical solutions of topologically stable vortex strings extending into the
bulk. We find evidence that the full UV expansion is dual to a point-like
boundary excitation.

##  1 Introduction

The advent of the Anti-de Sitter/Conformal Field Theory (AdS/CFT)
correspondence, as introduced by Maldacena Maldacena:1997re , revolutionized
our understanding of strongly coupled systems. This theoretical construct
bridges string theory in negatively curved AdS spacetimes with CFTs that
inhabit the spacetime boundary111 Here, the term ”boundary” refers to the
conformal boundary of the bulk space, distinct from the null boundary of
Minkowski space. In this context, signals within the bulk can reach the
boundary and return in a finite time frame. . Throughout the past two decades,
the AdS/CFT correspondence has been applied extensively within nuclear physics
and condensed matter, among other fields.

In the field of condensed matter physics, a rich spectrum of phenomena has
been modeled using the AdS/CFT framework, revealing gravitational duals for
each. Prominent examples include the quantum Hall effect Hartnoll:2007ai ;
Fujita:2009kw , the Nernst effect Hartnoll:2007ih , superconductivity
Hartnoll:2008kx , and Lifshitz scaling Kachru:2008yh .

The foundational work of Hartnoll et al. Hartnoll:2008kx ; Hartnoll:2008vx
laid the groundwork for holographic studies of superfluidity and
superconductivity. Their research demonstrated that, below a certain critical
temperature, a scalar field, which initially has a negligible expectation
value, develops a non-zero value, heralding the emergence of spontaneous
symmetry-breaking of a dual $U(1)$ symmetry. They particularly noted a sharp
increase in direct current (DC) conductivity at the critical temperature,
characteristic of a holographic superconductor.

Typically, the $(3+1)$-dimensional Abelian-Higgs model serves as the bulk
theory for such investigations. There is also much attention to the
understanding holographic dual of strongly coupled Bose-Einstein Condensates
(BECs). Regarding BECs, various studies have simulated the dynamics of
strongly coupled $U(1)$ physics with inhomogeneous phases Keranen:2009ss ;
Keranen:2009re , with multiple component scalars Yang:2020wes ; Yang:2019ibe ;
Yang:2022foe ; Kasamatsu:2003ktu , and under rotation Sonner:2009fk ;
Gregory:2014uca ; Srivastav:2023qof . Further studies have expanded on these
condensed matter phenomena of vortices Dias:2013bwa ; Albash:2009iq ;
Wittmer:2020mnm ; Wittmer:2021oxf  and lattice phases Su:2023vqa ; Xia:2019eje
. A central aspect in this line of research is the breaking of $U(1)$ symmetry
within certain boundary theories. This symmetry breaking, often facilitated by
a non-zero $U(1)$ chemical potential, dual to a gauge field in the bulk. The
$U(1)$ breaks at a critical temperature, resulting in a scalar condensate.

In this work, we present an alternative approach to the breaking of the global
$U(1)$ symmetry in the bulk. Our model differs from previous works by the
presence of a bulk $U(1)$ breaking scalar vacuum. We find that this vacuum
solution’s existence requires that the scalar’s mass squared, $m^{2}$, becomes
negative, proportional to the quartic coupling, $\lambda$, i.e.
$\lambda=-m^{2}L^{2}$, where $L$ is the radius of AdS. We will also probe two
bulk AdS geometries: AdS with a hard wall and AdS Black Brane. We find that
this scalar vacuum is stable against sourceless perturbations.

Our analysis explores topologically stable one-dimensional bulk vortices. We
examine single vortex and pair vortex configurations. For the pair of vortices
we simplify the physical model by only considering well-separated vortex
strings. We find evidence that the vortices are dual to a boundary field state
which has a localized non-zero scalar vacuum expectation value (VEV). The
candidate VEV condensates are found to center around vortex termination
points. However, we find that vortices severely warp the geometry at the
boundary; bulk vortices up to the boundary cannot be treated as probe fields.
This possibly indicates that bulk vortices can be treated as nucleation sites
for a phase transition and required us to impose a UV-cutoff in the bulk
coordinate.

Furthermore, we find numerical radial profiles of the vortex string pairs. We
find that vortex-vortex and vortex-antivortex pairs repel and attract,
respectively. Furthermore, we find a possible link with the near boundary
behavior of radial profile and the scalar VEV. We also find that these
vortices can have valid radial profiles only above a certain minimum
temperature.

This paper is organized in the following way. In Section 2, we will introduce
the bulk theory and derive the $U(1)$ symmetry breaking vacuum. We will also
will analyze the single vortex equation of motion for a vortex centered at
$x=y=0$ that ends on boundary. After, we will analyze the radial profiles for
well-separated vortex strings. In Section 3, we will cover analytical and
numerical results about the vortex strings. Finally, we will conclude in
Section 4

##  2 Analysis

Our bulk gravitational theory is global $\phi^{4}$ $U(1)$ scalar plus Einstein
Gravity with a negative cosmological constant. The background geometry is
taken to be asymptotically AdS. The bulk dimensions are $(3+1)$D. The action
takes the simple form of (1).

$\displaystyle\mathcal{S}=\mathcal{S}_{\mathrm{gravity}}+\mathcal{S}_{\mathrm{matter}}=$
$\displaystyle\int\sqrt{-g}\left(R-2\Lambda_{C}\right)$ (1) $\displaystyle-$
$\displaystyle\int\sqrt{-g}\left(g^{\mu\nu}\left(\partial_{\mu}\phi\right)\left(\partial_{\nu}\phi\right)^{\star}+V(|\phi|^{2})\right)$
$-\frac{1}{\sqrt{-g}}\partial_{\mu}\left(g^{\mu\nu}\sqrt{-g}\partial_{\nu}\phi\right)+V^{\prime}(|\phi|^{2})\phi=0$
(2)

$\phi$ is the complex scalar field. $V$ is the quartic potential such that

$V(|\phi|^{2})=\frac{\lambda}{2}\left(|\phi|^{2}\right)^{2}+m^{2}|\phi|^{2}=\frac{\lambda}{2}\left(|\phi|^{2}+\frac{m^{2}}{\lambda}\right)^{2}-\frac{\left(m^{2}\right)^{2}}{2\lambda}\,.$
(3)

There are several choices we could make for the analysis in this paper like in
terms of deforming the geometry. For this work we will restrict ourselves to
consider a simple, non-deformed asymptotically AdS geometry in which has a
metric of the form (4) with probe matter fields.

$g=\frac{L^{2}}{z^{2}}\left(-f(z)dt^{2}+\frac{1}{f(z)}dz^{2}+\delta_{2}\right)$
(4)

$\delta_{2}\equiv dx^{2}+dy^{2}$ is the 2D Euclidean metric.
$t,\,x,\,y\in\mathbb{R}$, and $z\in\mathbb{R}^{+}$. Furthermore, $f(z)$ is the
blackening factor. We consider two cases for $f(z)$. The first case covered is
$f(z)=1$, AdS. For AdS, we will artificially introduce a hard wall (HW). ”hard
wall” here means a surface were the scalar fields are set by hand to end. The
hard wall is a surface at points were $z=z_{\mathrm{h}}$. The second case is
$f(z)=1-z^{3}/z_{\mathrm{h}}^{3}$, AdS black brane; $z_{\mathrm{h}}$ is the
horizon radius. We find in Section 3.2, the radial profiles of the vortex
strings must obey Neumann boundary conditions on the horizon (and on the
boundary). That is to say, that vortices end on $z_{\mathrm{h}}$ and start on
the boundary.

The keen-eyed reader might have observed that the hard wall radius and the
horizon radius are denoted by the same symbol. Since they both function in the
same way to terminate the vortices and lead to similar vortex profiles, we
will keep the notation for simplicity.

###  2.1 Near Boundary Expansion

We would like to find vacuum solutions to (2) near conformal boundary such
that $\phi\sim z^{0}$. Assuming a real homogeneous solution,
$\left(\phi=z^{\beta}/L^{\beta+1}\right)$ near the boundary (so $f(z)\approx
1$) (5).

$-\beta\left(\beta-3\right)L^{-\beta-3}z^{\beta}+m^{2}L^{-\beta-1}z^{\beta}+L^{-3\beta-3}\lambda
z^{3\beta}=0$ (5)

For $z\to 0$, $\beta\left(\beta-3\right)=m^{2}L^{2}$ is asymptotically true if
$\beta>0$. Nonetheless, in this paper, we take another admissible asymptotic
solution. In this solution $\beta=0$. Namely, we take $\phi\sim z^{0}$, so
that

$L^{2}m^{2}=-\lambda\,.$ (6)

This latter solution implies that the $U(1)$ vacuum in the bulk preserves the
AdS asymptotics as long as $L^{2}m^{2}=-\lambda$. Further, there is a $U(1)$
symmetry-breaking phase. The vacuum value is
$|\phi_{\mathrm{vac}}|^{2}=-m^{2}/\lambda$ which conveniently simplifies to
$|\phi_{\mathrm{vac}}|^{2}=1/L^{2}$. We can absorb $L$ dependence by the
following rescaling of our fields as $g_{\mu\nu}\rightarrow g_{\mu\nu}/L^{2}$,
$\phi\rightarrow\phi L$, $m\rightarrow m/L$. Nevertheless, there is a drawback
to this ”constant” alternative boundary condition. The departure from the
standard boundary conditions thus requires more care in whether or not the
action/energies generated by such vacuum is renormalizable. The energy
momentum tensor is a linear combination of Lagrangian and square of
”$\partial_{\mu}$”. The solution is constant so all gradient terms vanish. The
source of any divergence would then have to come from the potential term in
the Lagrangian. In the $|\phi_{\mathrm{vac}}|^{2}=1$ vacuum, the current
potential is a constant $-\lambda/2$. Instead, we may consider the new non-
negative potential, shifted up by $\lambda/2$, since this alteration leaves
the equation of motion unchanged. Now the potential is zero in this vacuum,
leaving no contributions to the energy stress tensor. This again allows us to
treat the $\phi$ as a probe field. From this point onwards, the potential (3)
can be written as

$V(|\phi|^{2})=\frac{\lambda}{2}\left(|\phi|^{2}-1\right)^{2}\,.$ (7)

To test the linear stability of this vacuum we introduce perturbation of
$\phi$. Perturbing around this vacuum, we perturb $\phi$ as (8). We assume
that $|v|\ll|\phi_{\mathrm{vac}}|\equiv 1$ and $\eta\ll 1$.

$\phi=1+v+i\eta$ (8)

(8) can be plugged into the equations of motion and further expanded around
small $v$ near the boundary. For a homogeneous field, the near boundary
expansion will have a leading behavior of $\phi\rightarrow
1+v^{(0)}z^{\alpha}+i\eta^{(0)}z^{\gamma}$. The following equation gives an
indicial equation similar to a massive scalar field with a mass squared of
$2\lambda$. Also associated with the symmetry-breaking phase, the $\eta$ is a
massless scalar field. Such equivalent scalar perturbations are known to be
linearly stable Starinets:2002br ; Nunez:2003eq .

$\displaystyle\left(-\alpha\left(\alpha-3\right)+2\lambda\right)z^{\alpha}v^{(0)}+\mathrm{O}\left(\left(v^{(0)}\right)^{2}\right)$
$\displaystyle=0$ (9)
$\displaystyle-\gamma\left(\gamma-3\right)z^{\gamma}\eta^{(0)}+\mathrm{O}\left(\left(\eta^{(0)}\right)^{2}\right)$
$\displaystyle=0$

(9) implies two solutions for $\alpha$,

$\alpha=3/2\pm\sqrt{9/4+2\lambda}\,.$ (10)

Since $\lambda>0$, one of the solutions has diverging behavior for $\alpha$,
such that $\alpha<0$. This boundary behavior would violate a earlier
assumption that $|v|\ll|\phi_{\mathrm{vac}}|=1$. So in order to respect this
constraint, we need to take the positive solution in (10). The resulting
perturbation scalar field must not source any current on the boundary. $\eta$
does not have this restriction because both of its Frobenius indicial
exponents ($0$ or $3$) are non-negative. Now focusing on the scalar portion of
the action (1), it can now be written as

$\mathcal{S}_{\mathrm{scalar}}=-\int\sqrt{-g}\left(g^{\mu\nu}\left(\partial_{\mu}\phi\right)\left(\partial_{\nu}\phi\right)^{\star}+\frac{\lambda}{2}\left(|\phi|^{2}-1\right)^{2}\right)\,.$
(11)

###  2.2 Single Vortex

In this section we calculate the equation of motion for a single vortex
solution. Centering a vortex string at $x=y=0$ parallel to the $z$ axis, we
take the following ansatz.

$\phi=\mathit{h}(r,z)e^{ik\theta}$ (12)

where $k\in\mathbb{Z}$. Plugging in (12) into (2), we get the following PDE.

$-\frac{1}{\sqrt{-g}}\partial_{\mu}\left(g^{\mu\nu}\sqrt{-g}\partial_{\nu}\phi\right)+V^{\prime}(|\phi|^{2})\phi=0$
(13)
$z^{4}\partial_{z}\left(\frac{f}{z^{2}}\partial_{z}\mathit{h}\right)+\frac{z^{2}}{r}\partial_{r}\left(r\partial_{r}\mathit{h}\right)-\frac{z^{2}}{r^{2}}k^{2}\mathit{h}-\lambda\left(\mathit{h}^{2}-1\right)\mathit{h}=0$
(14)

(14) is a second order non-linear boundary value problem. We enforce boundary
conditions to make the solution to (14) well-defined. At $r=0$ we set the
Dirichlet $\mathit{h}(0,z)=0$. For $r=\infty$ we set a Dirichlet boundary
condition $\mathit{h}(\infty,z)=1$. At both ends of the $z$ domain we set
Neumann boundary conditions where
$\mathit{h}^{(0,1)}(r,0)=\mathit{h}^{(0,1)}(r,z_{\mathrm{h}})=0$. One might
think that a separation of variables could be applied here, but geometrical
factor $z^{2}$ obstructs such a solution. Instead of analytical solutions we
find a family of numerical solutions. For an example solution we solved for
$\lambda=1$ and $z_{\mathrm{h}}=10$ for both the AdS+HW and black brane case
Figure 1.

Figure 1: Vortex $\mathit{h}$ for the AdS with hard wall (left) and black
brane (right) case where $\lambda=1$ and $z_{\mathrm{h}}=10$.

Figure 2:  These are surface plot where $\mathit{h}(r,z)=1/2$. The black brane
(left) is more similar to a cone shape than the AdS plus hard wall which has a
more curvature apparently. Both plots where calculated with a numerical
solution where $\lambda=1$ and $z_{\mathrm{h}}=10$.

Even though, we found numerical solutions, they are similar to vortices of
different $\lambda$ at different $z$ cross-sections. The width of the vortex
is proportional to $z/\sqrt{\lambda}$. In Figure 2, the dependence of vortex
string width to bulk coordinate can be seen. In the other direction, for near
the boundary the vortex’s width shrink as if under a large $\lambda$ to a
cross-sectionally a point vortex (up to the resolution of the numerical
method). It appears that at the boundary the vortex solution is singular. It
is still not clear, what boundary state corresponds to static single vortex
solutions. Usually, the VEV of the scalar operator,
$\left\langle\mathcal{O}_{\phi}\right\rangle$, is related to subleading
behavior of the scalar field. For the free scalar field case, the
$\left\langle\mathcal{O}_{\phi}\right\rangle$ is proportional to the
subleading behavior of on-shell solution near the boundary. For a generic on-
shell scalar field, $\phi$, this looks like
$\phi\sim\phi^{(0)}z^{\Delta_{-}}+\mathrm{C}_{1}\left\langle\mathcal{O}_{\phi}\right\rangle
z^{\Delta_{+}}$. $\Delta_{\pm}$ and $\mathrm{C}_{1}$ are indicial exponents
and a constant respectively. A proxy of the scalar VEV,
$\left\langle\mathcal{O}_{\phi}\right\rangle$, is the value of the on-shell
bulk field on the boundary. Since the first derivative is set to zero via a
Neumann boundary condition, we can use the second derivative as a proxy for
$\left\langle\mathcal{O}_{\phi}\right\rangle$. This is plotted in Figure 3.
The plots suggest that a dual vortex is a vacuum disturbance in the boundary
theory. Possibly, in the full UV theory, vortices correspond to point
nucleation sites on the boundary. Such a finding corroborates with Section
2.1.

Figure 3:  Evaluated with an $\epsilon\approx 1.46$, these are plots of the
second derivative of $\mathit{h}$ with respect to $z$. Plots for the black
brane and for the AdS + HW case indicate the a tapering VEV for large lateral
distances from the core of the region where the bulk vortex approaches the
boundary.

To continue in this work, we will impose a cutoff around the center of
vortices on the boundary where $0<\epsilon^{2}<r^{2}+z^{2}$. Furthermore, we
are dealing with global vortices, so the tension/mass of the vortex has a log
divergence. 222 Interestingly enough for the black brane for a gauged vortex
profile $h\approx\tanh(r/z^{2})$, we find the finite part of the action is
proportional to $\int dtM/T$, where $M$ is the mass of the vortex.  Since the
width of the vortex is finite, we can treat them as vortex string
approximations with enough spacing in the transverse direction.

###  2.3 Vortex String Approximation

For a sufficiently large separation relative to the size of the vortices,
point-like approximations can be justified for a pair of vortices. In the
context of a higher-dimensional framework, such an object can be
conceptualized as a string that penetrates the bulk, originating from the
boundary. This is illustrated in Figure 4. This representation allows for a
simplified mathematical description, where a single vortex string can be
approximately described by the following equation:

$\phi_{i}=e^{\pm i\Theta_{i}}$ (15)

where, $\Theta_{i}$ denotes the angle measured around a point lateral to a
given holographic position, $z$. For a string, the central location from which
$\Theta_{i}$ is measured will typically depend on $z$. In the case of vortex-
vortex or vortex-antivortex interactions, a product of two instances of (15),
representing two distinct locations, serves as an approximation. Given the
translational and rotational symmetries along the transverse directions, it is
practical to position the vortices symmetrically around the origin and along
the $x$-axis, as detailed in (16)-(18).

$\Theta_{12}=\Theta_{1}+s\Theta_{2}$ (16) $\displaystyle\tan(\Theta_{1})$
$\displaystyle=\frac{y}{x-R(z)}$ (17) $\displaystyle\tan(\Theta_{2})$
$\displaystyle=\frac{y}{x+R(z)}$ (18)

(16) can thus be interpreted as the combined angle measured from two distinct
vortices located symmetrically at positions $(x,y)=(\pm R(z),0)$. $s=+1$ for
vortex-vortex, and $s=-1$ for vortex-antivortex.

Figure 4:  Here are two sketches of vortex strings (as dashed lines) in two
different geometries. The left diagram is anti-de Sitter with an hard wall.
The vortex string (and the scalar) field is artificially set to end at
$z=z_{\mathrm{h}}$. The right diagram is the anti-de Sitter black brane case.
Here, the vortex string like before ends on the black brane’s horizon at
$z=z_{\mathrm{h}}$. The field theory side of the duality is physics that takes
place at the spatially $2$D conformal boundary, $z=0$.

##  3 Results

This section is dedicated to presenting the numerical and analytical results
derived from the methodologies and theories discussed in the previous section.

###  3.1 Radial Profile Analysis

We would like to examine the interaction energy between two approximated
string-like vortices. To approximate this energy, we take the energy of the
full approximation with ${\phi_{12}:=\exp(i(\Theta_{1}+s\Theta_{2}))}$ (16)
and subtract the energy contributions of ${\phi_{1}:=\exp(i\Theta_{1})}$ (17)
and ${\phi_{2}:=\exp(is\Theta_{2})}$ (18) where $s=\pm 1$. These vortex
strings have an interaction potential as described by (20). In terms of
Lagrangians, this takes the form of (19).

$U:=-\int
dx^{3}(\mathcal{L}(\phi_{12})-\mathcal{L}(\phi_{2})-\mathcal{L}(\phi_{1}))$
(19)

Using the vortex string ansatz and the scalar Lagrangian (11), (19) can be
written as.

$U=s\int
dz\,dx^{2}\frac{1}{2}\sqrt{g}g^{ij}(\partial_{i}\Theta_{1})(\partial_{j}\Theta_{2})\,.$
(20)

The vortices are parameterized by (17) and (18)333 i,j,k,…are boundary
coordinate indices.. After integrating out the $x$ and $y$ directions in (20),
we obtain (21).

$U(2{R(0)})=\int dz\frac{\pi
s}{z^{2}}\left(2{V_{\partial\mathcal{M}}}(R)-f(z)R^{\prime}(z)^{2}\left({V_{\partial\mathcal{M}}}(R)+1\right)\right)$
(21)

where
${V_{\partial\mathcal{M}}}(R):=\log\left(\left(\Lambda^{2}/R^{2}+1\right)/4\right).$
${V_{\partial\mathcal{M}}}(R)$ is the well known potential between point
vortices in $(2+1)$D Smith:2008xx ; Shellard:1987bv ; Nakano:2008nas ;
Ovchinnikov:1998tge . $\Lambda$ is an IR-cutoff in the boundary radius
direction, $r$. This is needed since for global vortices, because there is a
log divergence. Note that $R$ is a function of $z$. To find the extremized
energy contributing configuration, we seek $R$ such that (21) is extremized.
The resulting Euler-Lagrange equation is (22).

$\displaystyle-R^{\prime}(z)\Big{(}R(z)\big{(}2f(z)-zf^{\prime}(z)\big{)}\big{(}R(z)^{2}+\Lambda^{2}\big{)}\times\big{(}{V_{\partial\mathcal{M}}}(R)+1\big{)}+\Lambda^{2}zf(z)R^{\prime}(z)\Big{)}$
(22)
$\displaystyle+zf(z)R(z)\big{(}R(z)^{2}+\Lambda^{2}\big{)}R^{\prime\prime}(z)\left({V_{\partial\mathcal{M}}}(R)+1\right)=2\Lambda^{2}z$

We find in Section 3.2 that $R$ must obey two Neumann boundary conditions:

$R^{\prime}(0)=R^{\prime}(z_{\mathrm{h}})=0\,.$ (23)

###  3.2 Neumann Boundary Conditions

We would like to understand the behavior of our $R$ fields near the boundary
and the horizon/hard wall. We first will expand $R$ around the boundary,
$z=0$, with a Taylor series (24), solving (22) order by order.

$R(z)=\sum_{i}\frac{1}{i!}R^{(i)}(0)z^{i}$ (24)

where $R^{(i)}(z)\equiv d^{i}R(z)/{dz}^{i}$. We plug (24) into (22) to solve
it order by order in $z$ with $R^{(i)}(0)$. At the leading order, we find
$R^{(1)}(0)=0$. This can be verified by the reader by looking at Appendix A.
At the next leading order $R^{(2)}(0)$ can be solved in terms of ${R(0)}$.
However, ${R^{(3)}(0)}$ cannot be determined by lower order coefficients. In
later sections, both ${R(0)}$ and ${R^{(3)}(0)}$ can be determined to find
solutions to the boundary value problem, (22).

We also perform a near horizon analysis for the black brane background. We
perform a Taylor series around $z=z_{\mathrm{h}}$ (25). Again, we solve the
equations of motion order by order in $z$ with (25).

$R(z)=\sum_{i}^{\infty}\frac{1}{i!}R^{(i)}(0)(z-z_{\mathrm{h}})^{i}$ (25)

We find $R^{(1)}(z_{\mathrm{h}})=0$ at leading order. Analogously, we also
enforce a Neumann boundary condition on the hard wall case at
$z=z_{\mathrm{h}}$.

Regardless of background, these two Neumann boundary conditions along with
(22) admit constant $R$ solutions for large $z_{\mathrm{h}}$. In either
background, a shooting method can be employed to solve this boundary value
problem. In short, the shooting method (more in Appendix B) involves setting
${R(0)}$ on the boundary and an initial ${R^{(3)}(0)}$; we use a root finding
algorithm to find ${R^{(3)}(0)}$, such that both Neumann boundary conditions
are satisfied. It is apparent that in the general case, the stated boundary
value problem is intractable analytically for the authors writing this paper.
In lieu of general analytical results, numerical results are presented in
Section 3.3.

#### Interaction Energy Integration

(21) is generically divergent. This is due to the log term but nevertheless
was to be expected because the single vortex has a divergent on-shell action.
To deal with this divergence, we need to expand UV cutoff such that
$z>\epsilon>0$ and also $(x\pm R)^{2}+y^{2}>\epsilon>0$ similar to Section
2.2. This UV cutoff, $\epsilon$, and IR cutoff, $\Lambda$ both limit the
length scale of these vortex strings.

#### Large $R(z)$ Limit

In the strict limit of large $R(z)$ take only leading order of the limit as
$R(z)\rightarrow\infty$ of (22). Taking the leading order term to vanish, we
find it is equivalent to

$\left(2f(z)-zf^{\prime}(z)\right)R^{\prime}(z)=zf(z)R^{\prime\prime}(z)\,.$
(26)

In practice this is equivalent to only considering
${V_{\partial\mathcal{M}}}(R)$. The general solution of (26) is readily found
to be

$R(z)=\mathrm{C}_{1}\int_{0}^{z}\frac{y^{2}dy}{f(y)}+{R(0)}\,.$ (27)

where $\mathrm{C}_{1}\in\mathbb{R}$. For the AdS $f(z)=1$,

$R(z)={R(0)}+\frac{{R^{(3)}(0)}}{6}z^{3}\,.$

For the AdS black brane $f(z)=1-z^{3}/z_{\mathrm{h}}^{3}$,

$R(z)={R(0)}-{R^{(3)}(0)}\frac{z_{\mathrm{h}}^{3}}{6}\ln\left(1-z^{3}/z_{\mathrm{h}}^{3}\right)\,.$

To satisfy the Neumann boundary condition at $z=z_{\mathrm{h}}$ for either
geometry for large ${R(0)}$, ${R^{(3)}(0)}$ must vanish. This implies that any
non-trivial deformations of $R$ is partly induced by effects that scale with
$R^{c}$ where $c<0$.

#### Vortex String Boundary Scalar Expansion

Naively, we can treat the scalar field of a vortex string as a free massless
scalar field and perform a near-boundary expansion to extract the boundary
scalar operator’s VEV, $\left\langle\mathcal{O}_{\phi}\right\rangle$, and
sourcing term. This approach is valid because the $R$ degree of freedom varies
the phase of $\phi$, and the modes corresponding to the phase are massless, as
discussed in Section 2.1. The near boundary purely imaginary homogeneous
perturbations will be shown to have the same asymptotic behavior: $z^{0}$ and
$z^{3}$. With the ansatz in (15) and a constant normalization, $\phi_{(0)}$,
the expansion can be written as:

$\displaystyle\phi$
$\displaystyle\sim\phi_{(0)}e^{i\Theta_{12}({R(0)})}\left(1+\frac{1}{2}iz^{2}R^{(2)}(0)\Theta_{12}^{\prime}({R(0)})+\frac{1}{6}i{R^{(3)}(0)}z^{3}\Theta_{12}^{\prime}({R(0)})\right)+O\left(z^{4}\right)$
(28)
$\displaystyle\sim\phi_{(0)}e^{i\Theta_{12}({R(0)})}\left(1+\frac{1}{2}iz^{2}R^{(2)}(0)\Theta_{12}^{\prime}({R(0)})+\mathrm{C}_{1}\left\langle\mathcal{O}_{\phi}\right\rangle
z^{3}\Theta_{12}^{\prime}({R(0)})\right)+O\left(z^{4}\right)$

where
$\Theta_{12}^{\prime}({R(0)})\equiv(d\Theta_{12}/d{R(0)})=sy/\left(y^{2}+\left(x+{R(0)}\right)^{2}\right)-y/\left(y^{2}+\left(x-{R(0)}\right)^{2}\right)$.
The normalization is necessary because, in the legitimate boundary action, we
need to be able to vary it with respect to a sourcing term, which could be the
leading order term of $\phi$. This leading order term is of constant order and
can be identified with $\phi$ evaluated at $z=0$, i.e.,
$\phi(0)=\phi_{(0)}e^{i\Theta_{12}}$.

For a free bulk massless scalar field, the boundary action takes the form
$S_{\text{boundary}}\propto\int_{\partial\mathcal{M}}\left({\phi}^{(0)}\right)^{2}{\phi}^{(3)}$,
where $\phi\sim{\phi}^{(0)}(1+{\phi}^{(3)}z^{3})$. Via the AdS/CFT
correspondence, we can identify the source term as ${\phi}^{(0)}$ and the VEV
as $\left\langle\mathcal{O}_{\phi}\right\rangle={\delta
S_{\text{boundary}}}/{\delta{\phi}^{(0)}}$. Thus, the third leading order
coefficient is identified as the VEV, with
$\left\langle\mathcal{O}_{\phi}\right\rangle$ being the VEV of the associated
boundary scalar operator and $\mathrm{C}_{1}$ being an arbitrary constant.

For a pair of vortex strings, the VEV can be written as

$\left\langle\mathcal{O}_{\phi}\right\rangle\propto{R^{(3)}(0)}\Theta_{12}^{\prime}({R(0)})={R^{(3)}(0)}\left(\frac{sy}{\left(y^{2}+\left(x+{R(0)}\right)^{2}\right)}-\frac{y}{\left(y^{2}+\left(x-{R(0)}\right)^{2}\right)}\right)\,.$
(29)

While $\left\langle\mathcal{O}_{\phi}\right\rangle$ might seem related, it is
fundamentally different from the vacuum expectation value (VEV) of a vortex.
Unlike a vortex VEV, $\left\langle\mathcal{O}_{\phi}\right\rangle$ does not
share two key properties: first, it has a non-zero at the exact location where
it should be zero for a vortex, and second, it does not exhibit symmetry
breaking behavior far away from the vortex (where the VEV typically has a non-
zero value). Moreover, $\left\langle\mathcal{O}_{\phi}\right\rangle$ vanishes
for large $r$ and is more like an inverted vortex VEV, as the nonzero values
of $\left\langle\mathcal{O}_{\phi}\right\rangle$ are localized.

In the full UV expansion with the UV cutoff, $z_{\mathrm{cutoff}}$, the large
energy from the vortices indicates that the back reaction of the geometry must
be accounted for. This, coupled with the fact that
$\left\langle\mathcal{O}_{\phi}\right\rangle$ is localized, provides evidence
that such holographic global vortices can induce a phase transition, with the
vortices themselves potentially acting as nucleation sites. This phase
transition is similar to a cosmological phase bubbles Grana:2021zvf ;
Mazumdar:2018dfl .

###  3.3 Numerical Results

In this section, we present the numerical results derived from the vortex
approximation discussed in Section 2.3. We have two sets of numerical data
sets. The first set of data is from the shooting method where $z_{\mathrm{h}}$
and ${R(0)}$ are chosen to fix $R(z)$ and ${R^{(3)}(0)}$. The shooting method
is described in Appendix B. From this data we derived two sets of plots. In
the first set of plots we plot ${R^{(3)}(0)}$ vs ${R(0)}$. For the second set
of plots are $U(2{R(0)})$ vs ${R(0)}$ where $U(2{R(0)})$ is (21). For the
reader to better understand found radial profiles, we have displayed a subset
of them. Figure 5 shows a sample profiles of $R(z)$ vs $z$ for a AdS plus hard
wall. One can see in Figure 5 that as ${R(0)}$ increases the radial profile
levels out. This is in agreement with the large $R(z)$ solutions (27) where
$R(z)$ flattens out for ${R(0)}\to\infty$. Overall radial profiles do not
differ noticeably between the spacetimes so, the black brane case will be
plotted for these two sets of plots but for the later set.

Figure 5:  For black brane + HW, this is a graph of sample radial profiles,
$R(z)/{R(0)}$ vs $z$. $\epsilon=1/1000$, $\Lambda=10000$, and
$z_{\mathrm{h}}=25$. Across the $x=y=0$ axis, there is a mirror image of $R$
that describes the radial profile of the other vortex string.

For the next numerical results, we present data to analyze the behavior of
$z_{\mathrm{h}}$ versus ${R(0)}$ and ${R^{(3)}(0)}$. The data used for this is
several values of ${R^{(3)}(0)}$ and a range of ${R(0)}$ values. With this
input data we solved for $z_{\mathrm{h}}$ and $R(z)$, solving for
$z_{\mathrm{h}}$ such that $R^{(1)}(z_{\mathrm{h}})=0$.

####  ${R^{(3)}(0)}$ vs ${R(0)}$

This is the first set of data and plots base off solutions to the boundary
value problem defined with (22) and (23), using the shooting method (explained
in Appendix B). We find solutions for a range of values for
$z_{\mathrm{h}}\in\\{25,50,75\\}$ and
${R(0)}/z_{\mathrm{h}}\in\\{1,{(1+1/20)},...,5\\}$. Given these two
quantities, we determine the third quantity, ${R^{(3)}(0)}$, via the
aforementioned shooting method. For both geometries, the plots of
${R^{(3)}(0)}$ vs ${R(0)}$ are displayed in Figure 6. There are a few notable
features of this data. First, ${R^{(3)}(0)}$ decreases for large ${R(0)}$.
This bending seems to come from the interaction between these two vortices.
The farther apart the vortex strings the lower the ${R^{(3)}(0)}$. This also
has the possible effect of lowering
$\left\langle\mathcal{O}_{\phi}\right\rangle$ contributed by both vortex
strings. Second, it is not shown but for small ${R(0)}$, there are larger
values of ${R^{(3)}(0)}$ up to a maximum around ${R(0)}\sim
3/10z_{\mathrm{h}}$. For smaller ${R(0)}$, the numerical methods used by
Mathematica break down, and no more solutions can be found. But, our numerical
routines fail to converge so $R^{(1)}(z_{\mathrm{h}})=0$. Because of this the
minimum probed value of ${R(0)}$ is $\sim z_{\mathrm{h}}$. Around these values
of $R(0)\approx z_{\mathrm{h}}$ we suspect that the vortex string
approximation breaks down.

Figure 6:  For both backgrounds, these are two plots of ${R^{(3)}(0)}$ vs
${R(0)}$. Black brane is to the left, and AdS plus HW is to the right.
$\epsilon=1/1000$ and $\Lambda=10000$ is used.

#### Potential vs ${R(0)}$

Probing the same values of $z_{\mathrm{h}}$ and ${R(0)}$, this paragraph
examines the potential ($U(2{R(0)})$) of the vortex string configurations vs
their separation at the boundary. For the vortex-vortex configuration, $s=1$,
the potential represents a repelling force. On the other hand, for the vortex-
antivortex configuration, $s=-1$, the potential represents an attractive
force. These findings agree with what is known about vortex strings
Manton:2004tk . The numerical values over several ${R(0)}$ can be seen in
Figure 7. A striking feature of the plots in Figure 7 and Figure 6 is that the
results, up to numerical error, are very similar across geometries. This
implies that the black brane case for studying probe vortices is sufficient to
understand the physics. Any back-reaction would then have to factor in the
geometry. This is in line with the fact that the scalar vacuum is the same for
both geometries.

Figure 7:  For both backgrounds, these are two plots of $U$ vs ${R(0)}$. Black
brane is to the left, and AdS plus HW is to the right. $\epsilon=1/1000$ and
$\Lambda=10000$ is used. Here the vortex-vortex configuration ($s=1$) is
taken. This potential results in an repulsive force. For the vortex-antivortex
configuration ($s=-1$), the resulting potentials are just the negated version
of above, $-U$. This potential results in an attractive force.

####  $z_{\mathrm{h}}$ vs ${R(0)}$

The goal is to investigate the behavior of $z_{\mathrm{h}}$ for various values
of ${R(0)}$ and ${R^{(3)}(0)}$. To achieve this, we solve for radius profiles
for several values of ${R(0)}$ and ${R^{(3)}(0)}$. For a chosen ${R(0)}$ and
${R^{(3)}(0)}$, we search for $z_{\mathrm{h}}$ such that
$R^{(1)}(z_{\mathrm{h}})=0$. There are three notable parameter regions.

  1. 1.

For ${R^{(3)}(0)}>\rho_{+}$, where $\rho_{+}$ is a small positive number, no
termination $z$ value was found. The radial profiles diverge by increasing
monotonically, and the minimum value of such profiles is the initial value,
${R(0)}$.

  2. 2.

For ${R^{(3)}(0)}<-\rho_{-}$, where $\rho_{-}$ is a small positive number, the
profiles fall towards the center line. The larger the absolute value of
${R^{(3)}(0)}$, the closer the profile ends to the boundary in the bulk.

  3. 3.

For $-\rho_{-}<{R^{(3)}(0)}<\rho_{+}$, we observe the behavior of both
solutions, with a minimum at $z_{\mathrm{h}}$. For ${R^{(3)}(0)}=0$,
$z_{\mathrm{h}}$ is maximized for some ${R(0)}$ greater than a certain
threshold. The previous radial profiles solved for with the shooting method
belong in this category.

Figure 8:  A figure that includes a sketch of the three cases of radial
profiles. For the first two cases the radial profiles do not end on the
brane/hard wall. Only the middle path of case three hits the the other side
and can end on a Neumann boundary condition.

Figure 8 illustrates the three categories. For the third case, we have
included a plot for several values of ${R^{(3)}(0)}$ and
${R(0)}\in\\{10,11,\ldots,200\\}$ in Figure 9. With similar reasons as the
previous two paragraphs, $z_{\mathrm{h}}=0$ was found for ${R(0)}$ regardless
of ${R^{(3)}(0)}$. We again suspect that the vortex string approximation
breaks down here. There are a few interesting observations about this
numerical data. First, for a given ${R(0)}$, there is a maximum
$z_{\mathrm{h}}$. In the black brane case, there is a temperature that is
inversely proportional to $z_{\mathrm{h}}$, $T\propto 1/z_{\mathrm{h}}$. Such
a maximum $z_{h}$ in the allowed ${R(0)}$ implies there is a minimum
temperature. For a large temperatures, vortices can only get so far before
their internal structure becomes relevant. The close they get more they bend
until they can no longer satisfy the Neumann boundary conditions. On the other
hand, the allowed region for a given ${R^{(3)}(0)}$ is to the right of the
curves in Figure 9. We have verified this by scanning more values of
${R^{(3)}(0)}$ for $-\rho_{-}<{R^{(3)}(0)}<\rho_{+}$. One can see that if the
${R^{(3)}(0)}=0$ line continues up for all ${R(0)}$, the minimum temperature
also decreases. If this trend were to continue for all ${R(0)}$ then for
${R(0)}\to\infty$, $T_{\mathrm{min}}\to 0$. This agrees with the derived large
${R(0)}$ solutions.

Figure 9:  For the ${R^{(3)}(0)}\sim 0$ case, this is a graph of $z_{h}$ for
several values of ${R(0)}$ and three values of ${R^{(3)}(0)}$. The geometry is
AdS + HW.

##  4 Conclusion

We analyzed a bulk global $\phi^{4}$ $U(1)$ scalar plus Einstein gravity
theory in two $(3+1)$-dimensional asymptotically Anti-de Sitter (AdS)
spacetimes: AdS plus hard wall and AdS black brane. The bulk scalar field
$\phi$ is governed by the equation of motion in (2) and a quartic potential
(3). Near the conformal boundary, we found a constant $\phi\sim z^{0}$
solution that preserves the AdS asymptotics and exhibits $U(1)$ symmetry
breaking in the bulk, as discussed in Section 2.1. This vacuum is linearly
stable against scalar perturbations that do not source boundary operators.
This novel boundary condition required the scalar mass to be related to the
quartic coupling as $m^{2}=-\lambda/L^{2}$.

We investigated vortex solutions in the bulk $U(1)$ breaking vacuum, Section
2.2. We numerically solved for the magnitude profile, $\mathit{h}$, of a
single vortex, centered at $x=y=0$, parallel to the $z$-axis. $\mathit{h}$
obeyed a 2nd order non-linear ODE (14). The presence of the the conformal
factor $z^{2}$ prohibited a separation by parts solution. Nevertheless, some
interesting conclusions can be made if the solution is modeled approximated to
be separable. We found its width increases with the bulk coordinate $z$,
corresponding to a decrease in the effective coupling $\lambda$. Roughly, the
conformal factor $z^{2}$ makes the effect coupling at a cross section of $z$
to be $\sim\lambda/z^{2}$, so the radius of the vortex should go like $\sim
z/\sqrt{\lambda}$. This behavior can be seen in Figure 2. We found that at the
boundary, the vortex became singular up to numerical error. This necessitated
the use of a UV cut off around the singular point on the boundary. We believe
that this is evidence that the bulk vortex is dual to a nucleation sites for a
change of phase on the boundary.

We also studied vortex string approximation as pairs of interacting vortex-
vortex or vortex-antivortex configurations, Section 2.3. We found that the
radial profiles must satisfy Neumann boundary conditions at their termination
points, and their unique solution requires specifying the initial location
${R(0)}$ and the third derivative ${R^{(3)}(0)}$, Section 3.2. The global
nature of the vortices introduces a log divergence in the energy integral,
necessitating UV and IR cutoffs around the boundary ($z\to\infty$) and
$r\to\infty$. In the limit of large vortex separation, analytical expressions
for the radial profiles were found, Section 3.1. Neumann boundary conditions
restricted these profiles to only flat profiles. This also implied that for
pure AdS (without the HW) the radial profile is constant. By analogy to a
massless scalar field, we found evidence that ${R^{(3)}(0)}$ proportional to a
scalar VEV, $\left\langle\mathcal{O}_{\phi}\right\rangle$, Section 3.1.

We found vortex string pairs for a given ${R(0)}$ and $z_{\mathrm{h}}$ and
output ${R^{(3)}(0)}$ in Section 3.3. Results confirmed the repulsive and
attractive behaviors of vortex-vortex and vortex-antivortex pairs,
respectively. Further, we found that for large distances the induced candidate
$\left\langle\mathcal{O}_{\phi}\right\rangle$ decreases with inter-vortex
string distance. We also found vortex string pairs with a given ${R(0)}$ and
${R^{(3)}(0)}$ where $z_{\mathrm{h}}$ is the output. For large enough
$\left|{R^{(3)}(0)}\right|$ we found that the string profile diverge from each
other for ${R^{(3)}(0)}>0$ and converge to the center ($x=y=0$) for
${R^{(3)}(0)}<0$. We also found a maximum $z_{\mathrm{h}}$ for small
$\left|{R^{(3)}(0)}\right|$ corresponding to a minimum temperature
$T_{\mathrm{min}}\propto 1/{R(0)}$ in the black brane case.

This study found that the UV cutoff limited what could be said about the
boundary theory. Nevertheless, in order to integrate to the boundary with
vortices their diverging self-energies must be accounted for as back-reaction
to the geometry. This new boundary geometry would likely differ from a
conformally flat geometry. One possible connection is that vortices are
conformal defects, studied in Dias:2013bwa . It would interesting to see if
the vortex string solutions found in this work could be nucleation sites in a
phase transition. Furthermore, the vortex string approximation breaks down for
small enough ${R(0)}$, requiring future work where the full vortex solution
needs to be found. The recent study by Chaffey et al. Chaffey:2023xmz
explored a similar setup. Their method of breaking $U(1)$ symmetry is with a
UV brane that has a quartic potential.

###### Acknowledgements.

The work of MA is supported by the Japan Society for the Promotion of Science
Postdoctoral Fellowship for Research in Japan (Short-term). The work of ME is
supported in part by JSPS KAKENHI No. JP22H01221 and the WPI program
”Sustainability with Knotted Chiral Meta Matter (SKCM2)” at Hiroshima
University.

##  Appendix A Equation of motion Expansion

This is the expansion of the equation of motion for (22) near the boundary
$z\approx 0$.

$\displaystyle\frac{4\pi
sR^{\prime}(0)\left(\log\left(\frac{1}{4}\left(\frac{\Lambda^{2}}{{R(0)}^{2}}+1\right)\right)+1\right)}{z^{3}}$
(30) $\displaystyle+\frac{2\pi
s\left(R^{\prime\prime}(0)\left(\log\left(\frac{1}{4}\left(\frac{\Lambda^{2}}{{R(0)}^{2}}+1\right)\right)+1\right)+\frac{\Lambda^{2}\left(2-3R^{\prime}(0)^{2}\right)}{{R(0)}\left({R(0)}^{2}+\Lambda^{2}\right)}\right)}{z^{2}}$
$\displaystyle+\frac{2\pi\Lambda^{2}sR^{\prime}(0)\left(\left(3{R(0)}^{2}+\Lambda^{2}\right)\left(R^{\prime}(0)^{2}-2\right)-2{R(0)}\left({R(0)}^{2}+\Lambda^{2}\right)R^{\prime\prime}(0)\right)}{{R(0)}^{2}z\left({R(0)}^{2}+\Lambda^{2}\right)^{2}}$
$\displaystyle-\frac{\pi
s}{3\left({R(0)}^{3}\left({R(0)}^{2}+\Lambda^{2}\right)^{3}\right)}\biggl{(}\Lambda^{6}\biggl{(}-f^{(3)}(0){R(0)}^{3}(\log(4)-1)R^{\prime}(0)$
$\displaystyle\quad+6{R(0)}R^{\prime\prime}(0)+2R^{\prime}(0)^{4}-3R^{\prime}(0)^{2}\left({R(0)}R^{\prime\prime}(0)+4\right)\biggr{)}$
$\displaystyle\quad+{R(0)}^{2}\biggl{(}R^{\prime}(0)\biggl{(}f^{(3)}(0){R(0)}\biggl{(}\Lambda^{6}\log\left(\frac{\Lambda^{2}}{{R(0)}^{2}}+1\right)$
$\displaystyle\quad+{R(0)}^{2}\left(3{R(0)}^{2}\Lambda^{2}+{R(0)}^{4}+3\Lambda^{4}\right)\left(\log\left(\frac{1}{4}\left(\frac{\Lambda^{2}}{{R(0)}^{2}}+1\right)\right)+1\right)\biggr{)}$
$\displaystyle\quad-2{R^{(3)}(0)}\Lambda^{2}\left({R(0)}^{2}+\Lambda^{2}\right)^{2}\biggr{)}$
$\displaystyle\quad+{R(0)}R^{(4)}(0)\left({R(0)}^{2}+\Lambda^{2}\right)^{3}\left(\log\left(\frac{1}{4}\left(\frac{\Lambda^{2}}{{R(0)}^{2}}+1\right)\right)+1\right)\biggr{)}$
$\displaystyle\quad+6{R(0)}^{2}\Lambda^{4}\left(4{R(0)}R^{\prime\prime}(0)+R^{\prime}(0)^{4}-2R^{\prime}(0)^{2}\left({R(0)}R^{\prime\prime}(0)+3\right)\right)$
$\displaystyle\quad+3{R(0)}^{4}\Lambda^{2}\left(6{R(0)}R^{\prime\prime}(0)+4R^{\prime}(0)^{4}-3R^{\prime}(0)^{2}\left({R(0)}R^{\prime\prime}(0)+8\right)\right)\biggr{)}$
$\displaystyle+\frac{\pi
sz}{6{R(0)}^{4}\left({R(0)}^{2}+\Lambda^{2}\right)^{4}}\biggl{(}{R(0)}^{4}\left({R(0)}^{2}+\Lambda^{2}\right)^{4}\left(4f^{(3)}(0)R^{\prime\prime}(0)+R^{(5)}(0)\right)\log\left(\frac{4{R(0)}^{2}}{{R(0)}^{2}+\Lambda^{2}}\right)$
$\displaystyle\quad-\left({R(0)}^{12}\left(4f^{(3)}(0)R^{\prime\prime}(0)+R^{(5)}(0)\right)\right)+2\Lambda^{2}\biggl{(}f^{(3)}(0){R(0)}^{3}\biggl{(}3\left({R(0)}^{2}+\Lambda^{2}\right)^{3}R^{\prime}(0)^{2}$
$\displaystyle\quad-2{R(0)}\Lambda^{6}R^{\prime\prime}(0)\biggr{)}+R^{\prime}(0)\biggl{(}-3{R(0)}^{2}\left({R(0)}^{2}+\Lambda^{2}\right)^{2}\left(3{R(0)}^{2}+\Lambda^{2}\right)R^{\prime\prime}(0)^{2}$
$\displaystyle\quad-12\left(4{R(0)}^{2}\Lambda^{4}+5{R(0)}^{4}\Lambda^{2}+10{R(0)}^{6}+\Lambda^{6}\right)R^{\prime}(0)^{2}$
$\displaystyle\quad+2{R(0)}\left({R(0)}^{2}+\Lambda^{2}\right)\left(3{R(0)}^{2}\Lambda^{2}+6{R(0)}^{4}+\Lambda^{4}\right)\left(R^{\prime}(0)^{2}+6\right)R^{\prime\prime}(0)\biggr{)}\biggr{)}$
$\displaystyle\quad-2{R(0)}^{4}\Lambda^{6}\biggl{(}{R(0)}\biggl{(}4\left(2f^{(3)}(0){R(0)}-3{R^{(3)}(0)}\right)R^{\prime\prime}(0)+2{R(0)}R^{(5)}(0)$
$\displaystyle\quad-9R^{(4)}(0)R^{\prime}(0)\biggr{)}+5{R^{(3)}(0)}\left(3R^{\prime}(0)^{2}+2\right)\biggr{)}$
$\displaystyle\quad-2{R(0)}^{6}\Lambda^{4}\biggl{(}3{R(0)}\biggl{(}4\left(f^{(3)}(0){R(0)}-{R^{(3)}(0)}\right)R^{\prime\prime}(0)+{R(0)}R^{(5)}(0)$
$\displaystyle\quad-3R^{(4)}(0)R^{\prime}(0)\biggr{)}+7{R^{(3)}(0)}\left(3R^{\prime}(0)^{2}+2\right)\biggr{)}$
$\displaystyle\quad-2{R(0)}^{8}\Lambda^{2}\biggl{(}{R(0)}\biggl{(}-4\left({R^{(3)}(0)}-2f^{(3)}(0){R(0)}\right)R^{\prime\prime}(0)+2{R(0)}R^{(5)}(0)$
$\displaystyle\quad-3R^{(4)}(0)R^{\prime}(0)\biggr{)}+3{R^{(3)}(0)}\left(3R^{\prime}(0)^{2}+2\right)\biggr{)}$
$\displaystyle\quad-{R(0)}^{2}\Lambda^{8}\biggl{(}{R(0)}\left({R(0)}R^{(5)}(0)-6R^{(4)}(0)R^{\prime}(0)\right)$
$\displaystyle\quad+{R^{(3)}(0)}\left(-8{R(0)}R^{\prime\prime}(0)+6R^{\prime}(0)^{2}+4\right)\biggr{)}\biggr{)}$
$\displaystyle+O\left(z^{2}\right)$

Assuming that $R^{\prime}(0)=0$, we get the following simplified expansion.

$\displaystyle\frac{2\pi
s\left(R^{\prime\prime}(0)\left(\log\left(\frac{1}{4}\left(\frac{\Lambda^{2}}{{R(0)}^{2}}+1\right)\right)+1\right)+\frac{2\Lambda^{2}}{{R(0)}\Lambda^{2}+{R(0)}^{3}}\right)}{z^{2}}$
(31) $\displaystyle+\frac{1}{3}\pi
s\biggl{(}R^{(4)}(0)\left(-\log\left(\frac{\Lambda^{2}}{{R(0)}^{2}}+1\right)-1+\log(4)\right)$
$\displaystyle\quad-\frac{6\left(3{R(0)}^{2}\Lambda^{2}+\Lambda^{4}\right)R^{\prime\prime}(0)}{{R(0)}^{2}\left({R(0)}^{2}+\Lambda^{2}\right)^{2}}\biggr{)}$
$\displaystyle+\frac{1}{6}\pi
sz\biggl{(}4f^{(3)}(0)(\log(4)-1)R^{\prime\prime}(0)-\left(4f^{(3)}(0)R^{\prime\prime}(0)+R^{(5)}(0)\right)\log\left(\frac{\Lambda^{2}}{{R(0)}^{2}}+1\right)$
$\displaystyle\quad+R^{(5)}(0)(\log(4)-1)+\frac{4{R^{(3)}(0)}\Lambda^{2}\left(2{R(0)}\left({R(0)}^{2}+\Lambda^{2}\right)R^{\prime\prime}(0)-3{R(0)}^{2}-\Lambda^{2}\right)}{{R(0)}^{2}\left({R(0)}^{2}+\Lambda^{2}\right)^{2}}\biggr{)}$
$\displaystyle+O\left(z^{2}\right)$

##  Appendix B Shooting Method

Figure 10: This figure is a sketch of the shooting method. The gray line is an
initial guess with fixed ${R(0)}$ and ${R^{(3)}(0)}$. These values determine
the near-boundary expansion for $0<z<\epsilon$. For $\epsilon<z<z_{h}$, $R(z)$
can be solved as an initial value problem with initial conditions
$R(\epsilon)$ and $R^{(1)}(\epsilon)$. The previous steps implicitly define a
function of ${R^{(3)}(0)}$ (with ${R(0)}$ and $z_{\mathrm{h}}$ fixed) that
outputs $R^{(1)}(z_{\mathrm{h}})$. A root-finding algorithm can be used to
find the root of this function.

In this section, we explain the shooting method employed to determine the
radial profiles. For further clarification, a graphic representation of the
shooting method is provided in Figure 10. Our focus is on the second-order
ordinary differential equation (22), with an independent variable $z$ and a
dependent variable $R(z)$. The dependent variable $R(z)$ is subject to Neumann
boundary conditions (23) at both ends of the $z$ domain,
$\left(0,z_{h}\right)$.

We tackle this problem via the shooting method starting at $z=0$. For our
boundary value problem, we have a singularity at $z=0$, so we can only
evaluate the equation near the boundary where $z=\epsilon\sim 0$. The shooting
method requires $R(\epsilon)$ and $R^{\prime}(\epsilon)$. We vary the
combination of $(R(\epsilon),R^{\prime}(\epsilon))$ such that the Neumann
boundary condition at $z=z_{h}$ is satisfied. However, our desired parameters
are ${R(0)}$ and ${R^{(3)}(0)}$. To obtain these, we rewrite
$\left(R(\epsilon),R^{\prime}(\epsilon)\right)$ in terms of
$\left({R(0)},{R^{(3)}(0)}\right)$ near-boundary expansion given as

$\displaystyle R(\epsilon)$
$\displaystyle\approx{R(0)}+\frac{\Lambda^{2}\epsilon^{2}}{{R(0)}\left({R(0)}^{2}+\Lambda^{2}\right)\left(-\log\left(\frac{\Lambda^{2}}{4{R(0)}^{2}}+\frac{1}{4}\right)-1\right)}+\frac{1}{6}{R^{(3)}(0)}\epsilon^{3}+\ldots\,,$
(32) $\displaystyle R^{\prime}(\epsilon)$
$\displaystyle\approx\frac{2\Lambda^{2}\epsilon}{{R(0)}\left({R(0)}^{2}+\Lambda^{2}\right)\left(-\log\left(\frac{\Lambda^{2}}{4{R(0)}^{2}}+\frac{1}{4}\right)-1\right)}+\frac{1}{2}{R^{(3)}(0)}\epsilon^{2}+\ldots\,.$

up to fifth order for our case.

With this expansion of $R$ near the boundary, we can evaluate $R(\epsilon)$
and $R^{(1)}(\epsilon)$ to be the initial conditions of an initial value
problem. From $z=\epsilon$, $R$ can be numerically integrated to
$z=z_{\mathrm{h}}$. With ${R(0)}$ fixed, we can use ${R^{(3)}(0)}$ as a
shooting parameter, varying such that Neumann boundary condition at
$z=z_{\mathrm{h}}$ is satisfied. The steps of the shooting method can be
summarized as follows:

  1. 1.

Develop a near-boundary expansion around $z=0$, parameterizing it with
variables ${R(0)}$ and ${R^{(3)}(0)}$.

  2. 2.

With the near-boundary expansion, evaluate $R(\epsilon)$ and
$R^{(1)}(\epsilon)$.

  3. 3.

Using $R(\epsilon)$ and $R^{(1)}(\epsilon)$, solve the equation of motion from
$z=\epsilon$ to the other end of the domain at ${z=z_{\mathrm{h}}>0}$.

  4. 4.

Employ a numerical solver to adjust the initial values ${R(0)}$ and
${R^{(3)}(0)}$ such that $R^{(1)}(z_{h})$ is zero.

## References

  * (1) J.M. Maldacena, _The Large N limit of superconformal field theories and supergravity_ , _Adv. Theor. Math. Phys._ 2 (1998) 231 [hep-th/9711200].
  * (2) S.A. Hartnoll and P. Kovtun, _Hall conductivity from dyonic black holes_ , _Phys. Rev. D_ 76 (2007) 066001 [0704.1160].
  * (3) M. Fujita, W. Li, S. Ryu and T. Takayanagi, _Fractional Quantum Hall Effect via Holography: Chern-Simons, Edge States, and Hierarchy_ , _JHEP_ 06 (2009) 066 [0901.0924].
  * (4) S.A. Hartnoll, P.K. Kovtun, M. Muller and S. Sachdev, _Theory of the Nernst effect near quantum phase transitions in condensed matter, and in dyonic black holes_ , _Phys. Rev. B_ 76 (2007) 144502 [0706.3215].
  * (5) S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, _Holographic Superconductors_ , _JHEP_ 12 (2008) 015 [0810.1563].
  * (6) S. Kachru, X. Liu and M. Mulligan, _Gravity duals of Lifshitz-like fixed points_ , _Phys. Rev. D_ 78 (2008) 106005 [0808.1725].
  * (7) S.A. Hartnoll, C.P. Herzog and G.T. Horowitz, _Building a Holographic Superconductor_ , _Phys. Rev. Lett._ 101 (2008) 031601 [0803.3295].
  * (8) V. Keranen, E. Keski-Vakkuri, S. Nowling and K.P. Yogendran, _Inhomogeneous Structures in Holographic Superfluids: I. Dark Solitons_ , _Phys. Rev. D_ 81 (2010) 126011 [0911.1866].
  * (9) V. Keranen, E. Keski-Vakkuri, S. Nowling and K.P. Yogendran, _Inhomogeneous Structures in Holographic Superfluids: II. Vortices_ , _Phys. Rev. D_ 81 (2010) 126012 [0912.4280].
  * (10) W.-C. Yang, C.-Y. Xia, M. Nitta and H.-B. Zeng, _Fractional and Integer Vortex Dynamics in Strongly Coupled Two-component Bose-Einstein Condensates from AdS/CFT Correspondence_ , _Phys. Rev. D_ 102 (2020) 046012 [2003.09423].
  * (11) W.-C. Yang, C.-Y. Xia, H.-B. Zeng and H.-Q. Zhang, _Phase Separation and Exotic Vortex Phases in a Two-Species Holographic Superfluid_ , _Eur. Phys. J. C_ 81 (2021) 21 [1907.01918].
  * (12) W.-C. Yang, C.-Y. Xia, H.-B. Zeng, M. Tsubota and J. Zaanen, _Motion of a superfluid vortex according to holographic quantum dissipation_ , _Phys. Rev. B_ 107 (2023) 144511 [2212.14488].
  * (13) K. Kasamatsu, M. Tsubota and M. Ueda, _Vortex phase diagram in rotating two-component bose-einstein condensates_ , _Physical Review Letters_ 91 (2003) .
  * (14) J. Sonner, _A Rotating Holographic Superconductor_ , _Phys. Rev. D_ 80 (2009) 084031 [0903.0627].
  * (15) R. Gregory, P.C. Gustainis, D. Kubizňák, R.B. Mann and D. Wills, _Vortex hair on AdS black holes_ , _JHEP_ 11 (2014) 010 [1405.6507].
  * (16) A. Srivastav and S. Gangopadhyay, _Vortices in a rotating holographic superfluid with Lifshitz scaling_ , _Phys. Rev. D_ 107 (2023) 086005 [2302.01030].
  * (17) O.J.C. Dias, G.T. Horowitz, N. Iqbal and J.E. Santos, _Vortices in holographic superfluids and superconductors as conformal defects_ , _JHEP_ 04 (2014) 096 [1311.3673].
  * (18) T. Albash and C.V. Johnson, _Vortex and Droplet Engineering in Holographic Superconductors_ , _Phys. Rev. D_ 80 (2009) 126009 [0906.1795].
  * (19) P. Wittmer, C.-M. Schmied, T. Gasenzer and C. Ewerz, _Vortex Motion Quantifies Strong Dissipation in a Holographic Superfluid_ , _Phys. Rev. Lett._ 127 (2021) 101601 [2011.12968].
  * (20) P. Wittmer, _Application of Black-Hole Physics to Vortex Dynamics in Superfluids_ , Ph.D. thesis, Heidelberg U., 2021.  10.11588/heidok.00029293.
  * (21) J.-H. Su, C.-Y. Xia, W.-C. Yang and H.-B. Zeng, _Vortex-Antivortex Lattices in a Holographic Superconductor_ , 2311.05856.
  * (22) C.-Y. Xia, H.-B. Zeng, H.-Q. Zhang, Z.-Y. Nie, Y. Tian and X. Li, _Vortex Lattice in a Rotating Holographic Superfluid_ , _Phys. Rev. D_ 100 (2019) 061901 [1904.10925].
  * (23) A.O. Starinets, _Quasinormal modes of near extremal black branes_ , _Phys. Rev. D_ 66 (2002) 124013 [hep-th/0207133].
  * (24) A. Nunez and A.O. Starinets, _AdS / CFT correspondence, quasinormal modes, and thermal correlators in N=4 SYM_ , _Phys. Rev. D_ 67 (2003) 124013 [hep-th/0302026].
  * (25) C.J. Pethick and H. Smith, _Bose–Einstein Condensation in Dilute Gases_ , Cambridge University Press, 2 ed. (2008).
  * (26) E.P.S. Shellard, _Cosmic String Interactions_ , _Nucl. Phys. B_ 283 (1987) 624.
  * (27) E. Nakano, M. Nitta and T. Matsuura, _Non-abelian strings in hot or dense qcd(cgc/lattice qcd/etc.,new frontiers in qcd 2008-fundamental problems in hot and/or dense matter-)_ , _Progress of Theoretical Physics Supplement_ (2008) 254.
  * (28) Y.N. Ovchinnikov and I.M. Sigal, _The ginzburg-landau equation iii. vortex dynamics_ , _Nonlinearity_ 11 (1998) 1277 .
  * (29) M. Graña and A. Herráez, _The Swampland Conjectures: A Bridge from Quantum Gravity to Particle Physics_ , _Universe_ 7 (2021) 273 [2107.00087].
  * (30) A. Mazumdar and G. White, _Review of cosmic phase transitions: their significance and experimental signatures_ , _Rept. Prog. Phys._ 82 (2019) 076901 [1811.01948].
  * (31) N.S. Manton and P. Sutcliffe, _Topological solitons_ , Cambridge Monographs on Mathematical Physics, Cambridge University Press (2004), 10.1017/CBO9780511617034.
  * (32) I. Chaffey, S. Fichet and P. Tanedo, _Holography of Broken U(1) Symmetry_ , 2309.00040.