# Novel vortices and the role of complex chemical potential in a rotating
holographic superfluid

Ankur Srivastav111<EMAIL_ADDRESS>Sunandan
<EMAIL_ADDRESS>
<EMAIL_ADDRESS>Department of Theoretical Sciences, S. N.
Bose National Centre for Basic Sciences,
Block-JD, Sector-III, Salt Lake City,
Kolkata 700106, India

###### Abstract

In this work, we have analytically devised novel vortex solutions in a
rotating holographic superfluid. To achieve this result, we have considered a
static disc at the $AdS$ boundary and let the superfluid rotate relative to
it. This idea has been numerically exploited in prdR  where formation of
vortices in such a setting was reported. We have found that these vortex
solutions are eigenfunctions of angular momentum. We have also shown that
vortices with higher winding numbers are associated with higher quantized
rotation of the superfluid. We have, then, analysed the equation of motion
along bulk AdS direction using Stürm-Liouville eigenvalue approach. A
surprising outcome of our study is that the chemical potential must be purely
imaginary. We have then observed that the winding number of the vortices
decreases with the increase in the imaginary chemical potential. We conclude
from this that imaginary chemical potential leads to less dissipation in such
holographic superfluids.

##  I Introduction

The emergence of gauge/gravity duality in the past decade has been
instrumental in our understanding on strongly correlated systems. This
connection between a $d$-dimensional gravity theory and $(d-1)$-dimensional
QFT has been applied to various physical systems ranging from early universe
cosmology to condensed matter systems like high-$T_{c}$ superconductors and
strongly coupled superfluids cph ; rgc , to name a few . The holographic
superconductors and superfluids have been studied in various spacetime
settings using numerical as well as analytical methods. Some crucial
properties associated with these phenomenon have been shown in the past few
years hhh ; hhh1 ; sah ; gr1 ; gr2 ; sg1 ; gg1 ; js ; assg ; asdgsg ; rbsg ;
gg2 ; gg3 ; pmchl . In particular, formation of vortex lattice in holographic
superconductors near second critical magnetic field has been shown mno ; prl ;
cvj1 ; cvj2 ; cvj3 . Also, it has been observed that vortices are formed if we
rotate a superfluid in a cylindical container. It is known from various
experiments that there are a variety of possible vortices in a superfluid
under rotation ovlet ; gev . Existence of such vortices is of prime interest
in a holographic superfluid model. Numerical studies leading to the existence
of such vortices in a rotating holographic superfluid have been carried out in
prdR ; xtzh . The study made use of the gauge/gravity duality to investigate
the dynamics of a strongly coupled superfluid in an uniformly rotating disk at
a finite temperature. As the angular velocity of the disk is increased above a
critical value, a vortex with quantized vorticity gets excited. With further
increase of the angular velocity, higher vortices are generated. In this
paper, we have analytically devised novel vortex solutions for a rotating
holographic superfluid model proposed in prdR . In our study, we consider that
there is a static disc of radius $R$ at the $AdS$ boundary and the superfluid
rotates relative to this disc. The superfluid being incompressible, demands no
flow along the radial direction and hence it is an equivalent description for
the alternate scenario where the superfluid is static in an uniformly rotating
disc. The vortex solutions that we have constructed enjoy circular symmetry in
the rotating disc of radius $R$ and each of these solutions are eigenfunctions
of the angular momentum. To obtain vortices, we have analysed this model very
near to the critical value of rotation $\Omega_{c}$, where superfluid vortex
state appears. Remarkably, the rotating superfluid also shows the step
transitions of the angular velocity observed in prdR  leading to the
excitation of vortices. Interestingly, we have also discovered a linear
relation between the winding number associated with these vortices and the
angular velocity of the rotating superfluid.
It is well established that such a simple holographic superfluid model is
parametrized by temperature and chemical potential hhh1 . If one keeps the
temperature fixed then there happens to be a phase transition at a critical
value of the chemical potential, $\mu_{c}$. Above this critical chemical
potential, the system is in superfluid phase. We have analysed the equation
along bulk AdS direction above $\mu_{c}$ but very close to it. To solve the
holographic system in the bulk direction, we have used a variational technique
known as Stürm-Liouville eigenvalue approach. From this analysis, we observe
that the chemical potential must be purely imaginary in order to get a
consistent solution. Further, it turns out that there is a decrease in the
winding numbers with an increase in the imaginary chemical potential.
Appearance of imaginary chemical potential have occured earlier in the
literature. A good motivation to work with imaginary chemical potential arises
in non-perturbative studies in quantum chromodynamics (QCD) carried out using
techniques of lattice gauge theory. The consequence of an imaginary potential
in QCD is the periodicity of the Roberge-Weiss (RW) phase transition rw . A
holographic understanding of this phase transition has been achieved on an
Euclidean spacetime set up in gasp ; jr . In our study, however, the need for
an imaginary chemical potential arises in a geometry whose signature is
Lorentzian. It would therefore be interesting to see whether the results in
the Euclidean set up would still be applicable when the signature of the
spacetime geometry is Lorentzian. At present we can only say it will since the
temporal component of the gauge field vanishes at the black hole horizon as it
happens in the Euclidean scenario. This would require further investigation
which we shall not carry out here.
In order to understand imaginary chemical potential in this holographic model,
we have further solved the time-dependent equations for the matter field. We
have been able to show that in this timedependent case, imaginary chemical
potential competes with the imaginary frequency, which is related to the
dissipation in the system. Hence, we conclude that increase in the imaginary
chemical potential leads to decrease in vortex number, which implies less
dissipation in the system. It should be noted that the complex chemical
potential has been related to dissipation in ck . Also, a holographic model
for the color superconductivity in QCD with imaginary chemical potential was
studied recently kg .

We have organized this paper in the following way. In section (II), we set up
the model for holographic superfluid in a static black hole background in
$AdS_{3+1}$ spacetime. In section (III), we have constructed vortex solutions,
near critical rotation in the rotating disc. Section (IV) deals with the
Stürm-Liouville eigenvalue analysis. Then in the last section (V) of this
paper, we have concluded and made some remarks on our results.

##  II The Holographic Superfluid

We start by writing down the metric for a static black hole in $AdS_{3+1}$
spacetime with Eddington-Finkelstein coordinates prdR ,

$\displaystyle
ds^{2}=\dfrac{\textit{l}^{2}}{u^{2}}[-f(u)dt^{2}-2dtdu+dr^{2}+r^{2}d\theta^{2}]$
(1)

where the blackening factor is given by,

$f(u)=\bigg{(}1-u^{3}\bigg{)}~{}.~{}$

Here l is the $AdS$ radius and $u$ is the bulk direction scaled in such a way
that $u=0$ is the AdS boundary and $u=1$ is the event horizon of the black
hole. The coordinates $(r,\theta)$ define the $2D$ flat disc. For convenience,
we take unit $AdS$ radius (that is, $l=1$) and the cosmological constant
$\Lambda=-3$. The Hawking temperature associated with the above black hole
geometry is given by $T=\dfrac{3}{4\pi}$.

We now consider a simple model for holographic superfluid on top of this
geometry. The action for the matter section in this model is given by,

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{S}=\dfrac{l^{2}}{2\kappa_{4}^{2}e^{2}}\int_{\mathcal{M}}d^{4}x~{}\mathcal{L}_{m}~{}.$
(2)

The matter Lagrangian density, $\mathcal{L}_{m}$, consists of a Maxwell field
and a complex scalar field minimally coupled to $A_{\mu}$. More precisely
$\mathcal{L}_{m}$ is given by following expression,

$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}\mathcal{L}_{m}=-\dfrac{1}{4}F^{\mu\nu}F_{\mu\nu}-|D\Psi|^{2}-m^{2}|\Psi|^{2}$
(3) $F_{\mu\nu}\equiv\partial_{[\mu}A_{\nu]}~{},~{}D\equiv(\nabla-ieA)~{}$

where $m$ is the mass of the scalar field while $e$ is its charge. Note that
we will be working in the probe limit. In this limit any backreaction of the
matter field in the metric is neglected. To achieve this limit, we shall
rescale $A_{\mu}\rightarrow\dfrac{A_{\mu}}{e}$ and
$\Psi\rightarrow\dfrac{\Psi}{e}$ and take the limit $e\rightarrow\infty$.
Mathematically, it is equivalent to setting $e=1$ in the action of our theory.
Now varying the action, $\mathcal{S}$, for $\Psi$ and $A_{\mu}$ we get the
following equations of motion for the matter and the gauge fields repectively,

$\displaystyle(D^{2}-m^{2})\Psi=0$ (4)
$\displaystyle\nabla_{\nu}F_{\mu}^{~{}\nu}=j_{\mu}$ (5)

where the bulk current is defined as,

$\displaystyle j_{\mu}\coloneqq
i\\{(D_{\mu}\Psi)^{\dagger}\Psi-\Psi(D_{\mu}\Psi)\\}~{}.$ (6)

We shall now assume that all the fields are stationary as our interest lies in
equilibrium analysis of the rotating superfluid system. Also we would be
working with the axial gauge, that is, $A_{u}=0$, in which case eq.(4) reduces
to the following equation,

$\displaystyle\\{\mathcal{D}(u)+\mathcal{D}(r)+\dfrac{1}{r^{2}}\mathcal{D}(\theta)\\}\Psi(u,r,\theta)=0$
(7)

where the segregated derivative operators are given as,

$\displaystyle\mathcal{D}(u)\equiv
u^{2}\partial_{u}\Big{(}\dfrac{f(u)}{u^{2}}\partial_{u}\Big{)}+iu^{2}\partial_{u}\Big{(}\dfrac{A_{t}}{u^{2}}\Big{)}+iA_{t}\partial_{u}-\dfrac{m^{2}}{u^{2}}$
$\displaystyle\mathcal{D}(r)\equiv\dfrac{1}{r}\partial_{r}(r\partial_{r})-\dfrac{i}{r}\partial_{r}(rA_{r})-iA_{r}\partial_{r}-A_{r}^{2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\mathcal{D}(\theta)\equiv\partial_{\theta}^{~{}2}-i(\partial_{\theta}A_{\theta}+A_{\theta}\partial_{\theta})-A_{\theta}^{~{}2}~{}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

##  III The Vortex Solution

Our interest is in the equilibrium state where vortices exist. So we define a
deviation parameter, $\epsilon$, from the critical rotation, $\Omega_{c}$, by
the following relation,

$\displaystyle\epsilon\coloneqq\dfrac{\Omega-\Omega_{c}}{\Omega_{c}}$ (8)

where $\Omega$ is the constant angular velocity of the disc. As argued in prdR
, one should notice that there is a relative velocity between the superfluid
and the disc. Hence, a static superfluid in a rotating disc is justly
represented by a rotating superfluid in a static disc. In this analysis, we
are visualizing the latter scenario. Now, in order to study this system very
near to $\Omega_{c}$, we series expand the matter field $\Psi$, the gauge
field $A_{\mu}$ and the bulk current $j_{\mu}$ with respect to $\epsilon$ in
the following manner mno ,

$\displaystyle\Psi(u,r,\theta)=\sqrt{\epsilon}\Big{(}\Psi_{1}(u,r,\theta)+\epsilon\Psi_{2}(u,r,\theta)+...\Big{)}$
(9) $\displaystyle
A_{\mu}(u,r,\theta)=\Big{(}A_{\mu}^{(0)}(u,r,\theta)+\epsilon
A_{\mu}^{(1)}(u,r,\theta)+...\Big{)}$ (10) $\displaystyle
j_{\mu}(u,r,\theta)=\epsilon\Big{(}j_{\mu}^{(0)}(u,r,\theta)+\epsilon
j_{\mu}^{(1)}(u,r,\theta)+...\Big{)}~{}.$ (11)

###  III.1 Zeroth order solutions near AdS boundary

The zeroth order solutions for gauge fields, in axial gauge, that generates
the critical rotation field and the chemical potential are given by following
relations,

$\displaystyle
A_{t}^{(0)}(u)=\mu(1-u),~{}~{}~{}A_{r}^{(0)}=0,~{}~{}~{}A_{\theta}^{(0)}(r)=\Omega
r^{2}~{}.$ (12)

Notice that $A_{r}^{(0)}=0$ restricts any superfluid flow in the radial
direction while $A_{\theta}^{(0)}$ allows the superfluid to rotate.
Considering these zeroth order solutions for gauge fields near the $AdS$
boundary, we may rewrite eq.(7) for lowest order in $\epsilon$, that is,
$\mathcal{O}(\sqrt{\epsilon})$, in the following form,

$\displaystyle\\{\mathcal{D}^{(0)}(u)+\mathcal{D}^{(0)}(r)+\dfrac{1}{r^{2}}\mathcal{D}^{(0)}(\theta)\\}\Psi_{1}(u,r,\theta)=0$
(13)

where the derivative operators become,

$\displaystyle\mathcal{D}^{(0)}(u)\equiv
u^{2}\partial_{u}\Big{(}\dfrac{f(u)}{u^{2}}\partial_{u}\Big{)}+iu^{2}\partial_{u}\Big{(}\dfrac{A_{t}^{(0)}}{u^{2}}\Big{)}+iA_{t}^{(0)}\partial_{u}-\dfrac{m^{2}}{u^{2}}$
$\displaystyle\mathcal{D}^{(0)}(r)\equiv\dfrac{1}{r}\partial_{r}(r\partial_{r})~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
$\displaystyle\mathcal{D}^{(0)}(\theta)\equiv\partial_{\theta}^{~{}2}-i(\partial_{\theta}A_{\theta}^{(0)}+A_{\theta}^{(0)}\partial_{\theta})-A_{\theta}^{(0)2}~{}.~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$

We now use the method of separation of variables to solve eq.(13) and write
$\Psi_{1}(u,r,\theta)$ as a function of $u$ and $(r,\theta)$ separately in the
following manner,

$\displaystyle\Psi_{1}(u,r,\theta)=\Phi(u)\xi(r,\theta)~{}.$ (14)

With the above separation of matter field, eq.(13) provides the following
separated equations,

$\displaystyle\mathcal{D}^{(0)}(u)\Phi(u)=\lambda\Phi(u)$ (15)
$\displaystyle\\{\mathcal{D}^{(0)}(r)+\dfrac{1}{r^{2}}\mathcal{D}^{(0)}(\theta)\\}\xi(r,\theta)=-\lambda\xi(r,\theta)$
(16)

where $\lambda$ is an unknown separation constant. Note that both eq.(s)(15,
16) are eigenvalue equations with eigenvalue $\lambda$. In the subsequent
discussion we shall proceed to determine $\lambda$.

###  III.2 Solution for vortex in the rotating superfluid

Given the $2D$ rotational symmetry, we may choose the following ansatz,

$\displaystyle\xi(r,\theta)=\eta_{p}(r)e^{ip\theta}$ (17)

where $p\in\mathcal{Z}$ for the single valuedness of the solution. However,
one should note that $\eta_{p}(r)$ must satisfy certain boundary conditions
for regularity at the boundaries. In our case, we would be working with the
Neumann boundary conditions at $r=0$ as well as at $r=R$, that is,

$\displaystyle\partial_{r}\eta_{p}|_{r=0}=0=\partial_{r}\eta_{p}|_{r=R}$ (18)

where $R$ is the radius of the disc boundary.
Now using the above ansatz in eq.(16), we get the following differential
equation to be solved under the boundary conditions defined above,

$\displaystyle\partial_{r}^{~{}2}\eta_{p}(r)+\dfrac{1}{r}\partial_{r}\eta_{p}(r)+\Big{\\{}\lambda-\Big{(}\dfrac{p}{r}-\Omega
r\Big{)}^{2}\Big{\\}}\eta_{p}(r)=0.$ (19)

To solve for $\eta_{p}(r)$, we consider the following ansatz,

$\displaystyle\eta_{p}(r)=F_{p}(r)e^{-\Omega r^{2}/2}~{}.$ (20)

Utilising this form of $\eta_{p}(r)$ given by eq.(20), eq.(19) takes the form,

$\partial_{r}^{~{}2}F_{p}(r)+\Big{(}\dfrac{1}{r}-2p\Omega\Big{)}\partial_{r}F_{p}(r)+\Big{(}\tilde{\lambda}-2\Omega-\dfrac{p^{2}}{r^{2}}\Big{)}F_{p}(r)=0.$
(21)

We now proceed to solve eq.(21) using the Frobenius series solution method. So
we consider that $F_{p}(r)$ is given by the following series,

$\displaystyle
F_{p}(r)=\sum_{n=0}^{\infty}a_{n}r^{n+k}~{}~{}~{}~{},~{}~{}~{}~{}(a_{0}\neq
0)$ (22)

with $k$ being an integer. The derivatives of the above series solution with
respect to $r$ are given by,

$\displaystyle\partial_{r}F_{p}(r)=\sum_{n=0}^{\infty}a_{n}(n+k)r^{n+k-1}$
(23)

and

$\displaystyle\partial_{r}^{2}F_{p}(r)=\sum_{n=0}^{\infty}a_{n}(n+k)(n+k-1)r^{n+k-2}~{}.$
(24)

Using eqs.(22, 23 and 24) in eq.(21), we find the following condition,

$\displaystyle\sum_{n=0}^{\infty}a_{n}\\{(n+k)^{2}-p^{2})\\}r^{n+k}+$
$\displaystyle\sum_{n=0}^{\infty}a_{n}\\{\lambda+2\Omega(p-1-n-k)\\}r^{n+k+2}=0~{}.$
(25)

This implies that coefficient for each order of $r$ should separately satisfy
eq.(25), that is,

$\displaystyle
r^{k}~{}~{}~{}:~{}~{}~{}~{}~{}a_{0}(k^{2}-p^{2})=0~{}~{}\Longrightarrow~{}~{}k=\pm
p~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$ $\displaystyle
r^{k+1}:~{}~{}~{}~{}~{}a_{1}((k+1)^{2}-p^{2})=0~{}~{}\Longrightarrow~{}~{}(k+1)=\pm
p~{}.$

From the above conditions we consider $k=p$ for the regularity of the
solutions at $r=0$ and this yields $a_{1}=0$. The condition $k=p$ implies that
$p$ is an integer. Similarly setting the coefficient for $r^{(k+n+2)}$ equal
to zero, we get the following recurrence relation,

$\displaystyle\dfrac{a_{n+2}}{a_{n}}=\dfrac{(\lambda-2\Omega(n+1))}{((n+2)^{2}+2p(n+2))}$
(26)

where we have already used the condition $k=p$. This recurrence relation
connects all the even coefficients with $a_{0}$ and all the odd coefficients
with $a_{1}$. Hence, we would get series solution for $F_{p}(r)$ with even
terms only. Now in order to have normalizable solutions, we must terminate
this series at some point, which determines $\lambda$ in terms of $\Omega$ and
$n$, that is,

$\displaystyle\lambda=2\Omega(n+1).$ (27)

The above relation implies that the eigenvalue $\lambda$ is quantized. With
this condition, the above series solution becomes a polynomial of order $n$.
Thus we can write the solution for $\eta_{p}(r)$ with an additional index
depicting the order of the polynomial as,

$\displaystyle\eta_{p,n}(r)=a_{0}e^{-\Omega r^{2}/2}F_{p,n}(r)$ (28)

where

$F_{p,n}(r)=r^{p}\big{(}1+\dfrac{a_{2}}{a_{0}}r^{2}+\dfrac{a_{4}}{a_{0}}r^{4}+...+\dfrac{a_{n}}{a_{0}}r^{n}\big{)}~{}.$

Let us now discuss the family of solutions with $n=0$. In this case,

$F_{p,0}(r)=r^{p}$

and hence,

$\displaystyle\eta_{p,0}(r)=a_{0}r^{p}e^{-\Omega
r^{2}/2}~{}~{}~{}~{};~{}~{}~{}(\lambda=2\Omega)$ (29)

This solution is subjected to the Neumann boundary conditions mentioned
earlier. This means the following first derivative of eq.(29) must vanish at
the disc boundaries,

$\displaystyle\partial_{r}\eta_{p,0}(r)=a_{0}r^{p-1}e^{-\Omega
r^{2}/2}(p-\Omega r^{2})~{}.$ (30)

Now the boundary condition at $r=0$ gives the following lower bound for $p$,

$\displaystyle\partial_{r}\eta_{p,0}(r)|_{r=0}=0~{}~{}~{}~{}\Longrightarrow~{}~{}~{}~{}p>1.$
(31)

Applying the boundary condition at the disc boundary at $r=R$ gives the
following linear relation between $p$ and $\Omega$,

$\displaystyle\partial_{r}\eta_{p,0}(r)|_{r=R}=0~{}~{}~{}~{}\Longrightarrow~{}~{}~{}~{}p=\Omega
R^{2}.$ (32)

Since $p$ is an integer, hence the above relation between $p$ and $\Omega$
implies a quantization of the angular velocity $\Omega$ and also a
quantization of the angular momenta in the rotating superfulid. Note that the
radius $R$ in the model is fixed. This implies that there is a linear relation
between $p$ and $\Omega$. This is a nice result that comes from our analysis.
Let us now consider the solution for $n=2$, which is given as,

$\displaystyle\eta_{p,2}(r)=a_{0}r^{p}e^{-\Omega
r^{2}/2}\Big{(}1-\dfrac{2\Omega}{(p+2)}r^{2}\Big{)}~{}~{};~{}(\lambda=6\Omega)~{}.$
(33)

For this solution, we have,

$\displaystyle\partial_{r}\eta_{p,2}(r)=a_{0}r^{p-1}e^{-\Omega
r^{2}/2}\Big{(}p-3\Omega r^{2}+\dfrac{2(\Omega r^{2})^{2}}{p+2}\Big{)}.$ (34)

In this case, the boundary condition at $r=0$ gives us the same lower bound
for $p$,

$\displaystyle\partial_{r}\eta_{p,2}(r)|_{r=0}=0~{}~{}~{}~{}\Longrightarrow~{}~{}~{}~{}p>1.$
(35)

However, the boundary condition at $r=R$ gives us the following condition,

$\displaystyle\partial_{r}\eta_{p,2}(r)|_{r=R}=0~{}\Longrightarrow~{}\Big{(}p-3\Omega
R^{2}+\dfrac{2(\Omega R^{2})^{2}}{p+2}\Big{)}=0.~{}~{}$ (36)

From this condition, we get,

$\displaystyle\Omega
R^{2}=3\dfrac{(p+2)}{4}\Big{(}1\pm\sqrt{1-\dfrac{8p}{9(p+2)}}\Big{)}.$ (37)

For $p>>2$, the above result again provides a linear relation between $p$ and
$\Omega$, that is, $\Omega R^{2}\sim p$.

Figure 1: Un-normalized lowest order ($n=0$) vortex solutions for different
winding numbers. (The value of R is set to be equal to 10).

##  IV Stürm-Liouville Eigenvalue Analysis

In this section, we shall solve eq.(15) using Stürm-Liouville eigenvalue
approach. We shall consider the analysis near the critical chemical potential
($\mu\sim\mu_{c}$) so that we may take the following ansatz for the gauge
fields near the AdS boundary 333$A_{t}^{(0)}(u)=\mu(1-u)\simeq\mu$ for
$u\rightarrow 0$. Note that $A_{t}^{(0)}(u)$ vanishes at the black hole
horizon $u=1$.,

$\displaystyle
A_{t}^{(0)}(u)=\mu,~{}~{}~{}A_{r}^{(0)}=0,~{}~{}~{}A_{\theta}^{(0)}(r)=\Omega
r^{2}~{}.$ (38)

For simplicity, we shall consider $m^{2}=-2,\Delta=1$. With these
considerations, eq.(15) reduces to the following equation,

$\displaystyle
u^{2}\partial_{u}\Big{(}\dfrac{1-u^{3}}{u^{2}}\partial_{u}\Phi(u)\Big{)}+iu^{2}\partial_{u}\Big{(}\dfrac{\mu}{u^{2}}\Phi(u)\Big{)}$
$\displaystyle+i\mu\partial_{u}\Phi(u)+\dfrac{2}{u^{2}}\phi=2\Omega\Phi(u)~{}.$
(39)

Notice that we have considered only the case for $n=0$ and hence
$\lambda=2\Omega$. We now simplify eq.(39) in the following form,

$\displaystyle(1-u^{3})\partial_{u}^{2}\Phi-\Big{(}u^{2}+\dfrac{2}{u}-2i\mu\Big{)}\partial_{u}\Phi$
$\displaystyle-\Big{(}2\Omega-\dfrac{2}{u^{2}}+\dfrac{2i\mu}{u}\Big{)}\Phi=0~{}.$
(40)

Near AdS boundary ($u\rightarrow 0$), we can write $\Phi(u)$ in the following
manner,

$\Phi(u)\simeq~{}<\mathcal{O}_{1}>u\Lambda(u)$

so that $\Lambda(u)$ is subjected to the boundary conditions given below,

$\Lambda(0)=1~{}~{};~{}~{}\partial_{u}\Lambda(0)=0~{}.$ (41)

Using this in eq.(40), we get an equation for $\Lambda$ as given below,

$\displaystyle(1-u^{3})\Lambda^{\prime\prime}-(3u^{2}-2i\mu)\Lambda^{\prime}-(u+2\Omega)\Lambda=0~{}$
(42)

where ′ denotes derivative with respect to $u$. Considering $\Lambda$ to be
real, eq.(42) implies that $\mu$ must be purely imaginary for eq.(42) to have
a consistent solution. So we have $Re(\mu)=0$, and set $Im(\mu)=\mu^{I}$. With
this, eq.(42) becomes,

$\displaystyle(1-u^{3})\Lambda^{\prime\prime}-(3u^{2}+2\mu^{I})\Lambda^{\prime}-(u+2\Omega)\Lambda=0~{}.$
(43)

In order to cast eq.(43) into Stürm-Liouville form, we multiply it with
integrating factor $R(u)$ given below,

$\displaystyle
R(u)=\Big{(}\dfrac{1-u}{\sqrt{1+u+u^{2}}}\Big{)}^{\dfrac{2\mu^{I}}{3}}exp\Big{(}-\dfrac{2\mu^{I}}{\sqrt{3}}\arctan(\dfrac{1+2u}{\sqrt{3}})\Big{)}~{}.$

With this, eq.(43) can be put into Stürm-Liouville form as given below,

$\displaystyle(P(u)\Lambda^{\prime}(u))^{\prime}+Q(u)\Lambda(u)+\Omega
S(u)\Lambda(u)=0~{}$ (44)

where

$\displaystyle P(u)=(1-u^{3})R(u)$ $\displaystyle
Q(u)=-uR(u)~{}~{}~{}~{}~{}~{}$ $\displaystyle
S(u)=2R(u)~{}.~{}~{}~{}~{}~{}~{}~{}$

Now the eigenvalue $\Omega$ is given by the following integral,

$\displaystyle\Omega=\dfrac{\int_{0}^{1}du(P(u)(\Lambda^{\prime}(u))^{2}-Q(u)\Lambda^{2}(u))}{\int_{0}^{1}duS(u)\Lambda^{2}(u)}~{}.$
(45)

In order to proceed ahead, we take a trial function for $\Lambda(u)$ that
satisfies the given boundary conditions, that is,
$\Lambda(0)=1,~{}\partial_{u}\Lambda(0)=0~{}.$ We assume the following trial
function,

$\Lambda_{\alpha}(u)=(1-\alpha u^{2})~{}.$

With this trial function, we have the following equation to determine
$\Omega_{\alpha}$,

$\displaystyle\Omega_{\alpha}=\dfrac{\int_{0}^{1}du(P(u)(\Lambda_{\alpha}^{\prime}(u))^{2}-Q(u)\Lambda_{\alpha}^{2}(u))}{\int_{0}^{1}duS(u)\Lambda_{\alpha}^{2}(u)}~{}.$
(46)

In order to compute eq.(46), we approximate $R(u)$ for $u\rightarrow 0$ in the
following manner,

$\displaystyle
R(u)\simeq\Big{(}1-\dfrac{2\mu^{I}}{\sqrt{3}}\arctan(\dfrac{1+2u}{\sqrt{3}})\Big{)}~{}.$
(47)

Using eq.(47) into eq.(46), we find the following equation for
$\Omega_{\alpha}$,

$\displaystyle\Omega_{\alpha}=~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(48)
$\displaystyle\dfrac{\int_{0}^{1}du\Big{(}1-\dfrac{2\mu^{I}}{\sqrt{3}}\arctan(\dfrac{1+2u}{\sqrt{3}})\Big{)}\big{(}u+4\alpha^{2}u^{2}-2\alpha
u^{3}-3\alpha^{2}u^{5}\big{)}}{\int_{0}^{1}du\Big{(}1-\dfrac{2\mu^{I}}{\sqrt{3}}\arctan(\dfrac{1+2u}{\sqrt{3}})\Big{)}\big{(}1+\alpha^{2}u^{4}-2\alpha
u^{2}\big{)}}~{}.$

We now need to extremize $\Omega_{\alpha}$ with respect to $\alpha$. For a fix
value of $\mu^{I}$, it turns out that there are two values of $\alpha$ which
extremize eq.(48). To understand the qualitative role of $\mu^{I}$, we have
calculated these extremized values of $\Omega_{\alpha}$ for a range of values
of $\mu^{I}$. Extremized values of $\Omega_{\alpha}$, corresponding to both
values of $\alpha$, for $\mu^{I}$ between $4.0$ and $4.5$ are shown in Fig.(2)
and Fig.(3).

Figure 2: $\Omega$ vs $\mu^{I}$ for lowest order ($n=0$) vortex solutions for
first values of $\alpha$ that extremize $\Omega_{\alpha}$ in eigenvalue
equation (48). Figure 3: $\Omega$ vs $\mu^{I}$ for lowest order ($n=0$) vortex
solutions for second values of $\alpha$ that extremize $\Omega_{\alpha}$ in
eigenvalue equation (48).

These figures show a remarkable trend, in both the cases extremized values of
$\Omega$ consistently decreases with an increase in the value of imaginary
chemical potential. Now some subtle observations are in order here. As we have
shown in section (III) that these $\Omega$ are quantized with the following
relation,

$\Omega=\dfrac{p}{R^{2}}~{}$

where $R$ is the radius of the disc. This relation in conjuction with Fig.(2)
and Fig.(3) implies that for a disc with a fixed radius $R$, there is a
decrease in the winding numbers as the imaginary chemical potential rises.
This seems to be an interesting observation from holographic point of view. In
order to better understand this result, we have further considered the
timedependent terms in the equation of motions given by eq.(7). The
corresponding time-dependent equation is given as,

$\displaystyle\\{\mathcal{D}(u)+\mathcal{D}(r)+\dfrac{1}{r^{2}}\mathcal{D}(\theta)-2\partial_{u}\partial_{t}+\dfrac{2}{u}\partial_{t}\\}\Psi(t,u,r,\theta)=0.~{}~{}$
(49)

Linearizing this equation with the following form of $\delta\Psi$ and $\delta
A_{\mu}$ along with the boundary conditions expressed before,

$\delta\Psi=p(u,r)e^{i\omega t+in\theta}~{}~{};~{}~{}\delta
A_{\mu}=a_{t}(u,r)e^{i\omega t+in\theta}$

gives the following equations after separation of variables for
$p(u,r)=\Phi(u)\eta_{p}(r)$,

$\displaystyle(1-u^{3})\partial_{u}^{2}\Phi-\Big{(}u^{2}+\dfrac{2}{u}-2i(\mu-\omega)\Big{)}\partial_{u}\Phi$
$\displaystyle-\Big{(}-\dfrac{2}{u^{2}}+\dfrac{2i(\mu-\omega)}{u}\Big{)}\Phi=\lambda\Phi~{}.$
(50)
$\displaystyle\partial_{r}^{~{}2}\eta_{p}(r)+\dfrac{1}{r}\partial_{r}\eta_{p}(r)+\Big{\\{}\lambda-\Big{(}\dfrac{p}{r}-\Omega
r\Big{)}^{2}\Big{\\}}\eta_{p}(r)=0.$ (51)

One should notice that these are similar equations that we have found for
stationary field case in sections (III) and (IV). The only difference is that
in eq.(50), $\mu$ is now replaced with $(\mu-\omega)$. This difference
immediately points towards a connection between imaginary chemical potential
and imaginary part of the frequency, $\omega$. As it is well known that
imaginary part of the frequency, $\omega$, implies dissipation in the system,
hence we may attach a similar meaning to $\mu^{I}$. We observe from Fig.(2)
and Fig.(3) that there is a decrease in the number of vortices with a rise in
the value of the imaginary chemical potential. On the other hand, from
Fig.(4), we observe that with increase in $\omega$, $\Omega$ increases which
means that the number of vortices increases. Now increase in the vortex number
can be understood as an increase of dissipation in the system pmchl . Hence,
the presence of both the imaginary chemical potential,‘$\mu^{i}$’ and the
frequency ‘$\omega$’ of the quasi-normal modes reduces the dissipation in the
system.

Figure 4: $\Omega$ vs $\omega$ for lowest order ($n=0$) vortex solutions for
first values of $\alpha$ that extremize $\Omega_{\alpha}$ in eigenvalue
equation obtained by eq.(52). (see Appendix)

##  V Conclusion and Remarks

In this work, we have holographically devised vortex solutions with different
winding numbers in a rotating superfluid. These solutions may be interpreted
as vortices placed at the centre of the disc at $r=0$. Our analysis shows that
$p=\Omega R^{2}$ is an exact condition for $n=0$ case while it is true for
$p>>2$ for higher order solutions, that is, $n\neq 0$. This linear relation
between the winding number, $p$, and the angular velocity, $\Omega$, seems to
be an universal feature of such vortices at least for large $p$. It is to be
noted that due to the Neumann boundary condition at $r=0$, the vortex solution
with winding number $p=1$ is absent in this model. However, if one considers
the Dirichlet boundary condition, at $r=0$, instead of Neumann boundary
condition, then even solutions with winding number $p=1$ are allowed. In
Fig.(1), we have shown some vortex solutions with different winding numbers.
We have further solved the bulk equation using Stürm-Liouville eigenvalue
approach and have observed that the chemical potential must be purely
imaginary. A relation between winding numbers associated with the vortices and
the imaginary chemical potential for the specific case of lowest order ($n=0$)
vortices have been found. We have given a novel interpretation to this
relation in terms of reduction in the number of vortices in rotating
holographic superfluids, with increase in the imaginary chemical potential,
which in turn implies reduction of the dissipation in the system. As a final
remark, we would like to emphasize that the results in this work have been
obtained analytically making use of the gauge/gravity duality and has similar
features to those found numerically.

Acknowledgements: AS would like to thank Neeraj Kumar for some fruitful
discussions. AS would also like to acknowledge anonymous referee for critical
comments, which led us to some interesting findings.

## References

  * (1) Chuan-Yin Xia et.al., Phys. Rev. D 100, 061901(R) (2019)
  * (2) C. P. Herzog, 2009 J. Phys. A: Math. Theor. 42 343001
  * (3) R. Cai, L. Li et al. Sci. China Phys. Mech. Astron. 58, 1–46 (2015)
  * (4) S. A. Hartnoll, C. P. Herzog, G. T. Horowitz, Phys. Rev. Lett. 101, 031601.
  * (5) S. A. Hartnoll, C. P. Herzog, G. T. Horowitz, JHEP12(2008)015
  * (6) S. A. Hartnoll, 2009 Class. Quantum Grav. 26 224002
  * (7) S. Gangopadhyay, D. Roychowdhury, J. High Energ. Phys. (2012) 2012: 2.
  * (8) S. Gangopadhyay, D. Roychowdhury, J. High Energ. Phys. (2012) 2012: 156.
  * (9) S. Gangopadhyay, Physics Letters B Volume 724, Issues 1–3, 9 July 2013, Pages 176-181
  * (10) D. Ghorai, S. Gangopadhyay, Eur. Phys. J. C (2016) 76: 146.
  * (11) J. Sonner, Phys. Rev. D 80, 084031
  * (12) A. Srivastav, S. Gangopadhyay, Eur. Phys. J. C 79, 340 (2019).
  * (13) A. Srivastav, D. Ghorai, S. Gangopadhyay, Eur. Phys. J. C 80, 219 (2020).
  * (14) R. Banerjee, S. Gangopadhyay et.al. Phys. Rev. D 87, 104001
  * (15) D. Ghorai, S. Gangopadhyay Eur. Phys. J. C 76, 146 (2016)
  * (16) D. Ghorai, S. Gangopadhyay EPL 118, 31001 (2017)
  * (17) P.M. Chesler, H. Liu, A. Adam Science 341 (6144) 368
  * (18) K. Maeda, M. Natsuume, T. Okamura, Phys. Rev. D 81, 026002 (2010)
  * (19) M. Montull, A. Pomarol, P. J. Silva Phys. Rev. Lett. 103, 091601 (2009)
  * (20) T. Albash and C. V. Johnson, JHEP09 121 (2008)
  * (21) T. Albash and C. V. Johnson,, Phys. Rev. D 80, 126009 (2009)
  * (22) T. Albash and C. V. Johnson,, arXiv:0906.0519v1
  * (23) O. V. Lounasmaa, E. Thuneberg Proc. Natl. Acad. Sci. USA 96, 7760-7767 (1999)
  * (24) G. E. Volovik Physics-Uspekhi 58 (9) 897-905 (2015)
  * (25) X. Li, Y. Tian, H. Zhang, J. High Energ. Phys. 2020, 104 (2020).
  * (26) A. Roberge and N. Weiss Nucl. Phys. B275, 734 (1986)
  * (27) G. Aarts, S. P. Kumar, and J. Rafferty J. High Energy Phys. 07 (2010) 056.
  * (28) J. Rafferty J. High Energy Phys. 09 (2011) 087.
  * (29) G. E. Cragg and A. K. KermanPhys. Rev. Lett. 94, 190402
  * (30) K. Ghoroku et.al.Phys. Rev. D 102, 046003

##  VI Appendix

If we consider $\mu=0$ in eq.(50), then we get

$\displaystyle(1-u^{3})\partial_{u}^{2}\Phi-\Big{(}u^{2}+\dfrac{2}{u}+2i\omega\Big{)}\partial_{u}\Phi$
$\displaystyle-\Big{(}-\dfrac{2}{u^{2}}-\dfrac{2i\omega}{u}\Big{)}\Phi=\lambda\Phi~{}.$
(52)

Now comparing eq.(52) with eq.(40), we find that these two equations are
similar to each other with the difference of sign in $\mu$ and $\omega$. Now
using the Stürm-Liouville eigenvalue approach, we can solve eq.(52). The
resulting behaviour between $\Omega$ and $\omega$ is given in Fig.(4). This
figure shows that vortex number increases with the increase of quasi-normal
frequency, $\omega$. This implies that dissipation of the system increases
with increase in $\omega$.