[1,2]Piotr Magierski

These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

These authors contributed equally to this work.

[1]Faculty of Physics, Warsaw University of Technology, ulica Koszykowa 75,
Warsaw, 00-662, Poland

2]Physics Department, University of Washington, 3910 15th Ave. NE, Seattle, WA
98195-1560, USA

3]Institute of Physics, Polish Academy of Sciences, Aleja Lotnikow 32/46,
Warsaw, 02-668, Poland

# Quantum vortices in fermionic superfluids: from ultracold atoms to neutron
stars.

<EMAIL_ADDRESS>Andrea Barresi<EMAIL_ADDRESS>Andrzej Makowski<EMAIL_ADDRESS>Daniel Pęcak
<EMAIL_ADDRESS>Gabriel Wlazłowski<EMAIL_ADDRESS>*  [
[

###### Abstract

Superfluid dilute neutron matter and ultracold gas, close to the unitary
regime, exhibit several similarities. Therefore, to a certain extent,
fermionic ultracold gases may serve as emulators of dilute neutron matter,
which forms the inner crust of neutron stars and is not directly accessed
experimentally. Quantum vortices are one of the most significant properties of
neutron superfluid, essential for comprehending neutron stars’ dynamics. The
structure and dynamics of quantum vortices as a function of pairing
correlations’ strength are being investigated experimentally and theoretically
in ultracold gases. Certain aspects of these studies are relevant to neutron
stars. We provide an overview of the characteristics of quantum vortices in
s-wave-type fermionic and electrically neutral superfluids. The main focus is
on the dynamics of fermionic vortices and their intrinsic structure.

###### keywords:

fermionic superfluid, quantum vortices, vortex structure, vortex dynamics

##  1 Introduction

Pairing correlations in nuclear systems are essential for describing both
static and dynamic properties of nuclei. Such phenomena as odd-even mass
difference or back bending of moments of inertia of atomic nuclei cannot be
explained without invoking the concept of pair correlations (see [1, 2, 3] and
references threrein). However, the precise extraction of the pairing magnitude
is difficult due to the finite size of the system. The information about
pairing gap is obscured by polarization effects associated with adding or
substracting a single nucleon (see eg. [4] for discussion of the interplay
between pairing and mean-field effects in the odd-even mass staggering). For
most applications, it is sufficient to consider one particular feature of
pairing correlations: the presence of the energy gap at the Fermi surface.
Various manifestations of superfluidity can be traced back to this single
quantity, usually treated as a parameter of mean-field models (see e.g. [5]).
Therefore, it is still an open question whether pairing in nuclear systems can
be treated as a condensate of pairs, giving rise to the superfluid phase, or
it is merely an incoherent pair correlation. Clearly, the condition of
nonvanishing off-diagonal long-range order cannot be applied to atomic nuclei
due to their finite size. However, if a superfluid phase exists, it has to be
seen in phenomena requiring a well-defined phase factor of the pairing field.
These involve analogs of the Josephson effect [6, 7, 8, 9], both in DC and AC
realizations, and solitonic excitations [10, 11]. Manifestations of these
effects were reported to be seen in nuclear collisions through enhanced
neutron pair transfer [12, 13, 14], photon radiation of particular energy [15,
16, 17], or an increase of the barrier for capture of colliding nuclei [18].

On the other hand, even stronger pairing correlations are predicted to be
present in nuclear matter at subnuclear densities [19]. Unfortunately, dilute
nuclear, particularly neutron matter, cannot be directly accessed
experimentally. They form outer layers of neutron stars — the so-called inner
crust [20]. However, since the system is of macroscopic size and is
characterized by strong pairing correlations, the superfluid phase (of s-wave
type) is likely to be formed [21, 22, 23, 24]. Moreover, the dynamics of a
rotating star raises the natural question concerning the role of quantum
vortices in the dynamics of the crust, especially in the context of observed
glitch phenomenon [25, 26]. The presence of quantum vortices, which require a
particular configuration of pairing field, is possible only under the
assumption that the neutron matter is superfluid.

The properties of vortices in neutron stars can be modeled using Density
Functional Theory (DFT) based on the properties of known nuclear systems [27,
28]. However, since neutron matter at subnuclear densities is relatively close
to the so-called unitary regime, one can expect that the properties of
vortices are close to those observed in ultracold atomic gases. Therefore,
theoretical studies confronted with experiments on ultracold atomic gases can
help provide constraints for the theory of quantum vortices in neutron matter
[29]. Fermionic ultracold gases provide a particularly useful playground to
investigate properties of vortices [30], since they provide an example of a
system that is electrically neutral, contrary to metal superconductors, which
is close to the conditions in the inner crust (proton fraction is small and
localized in impurities). Therefore, all effects related to interactions with
magnetic and electric fields can be ignored.

In this review, we summarize the main characteristics of vortices and their
dynamics predicted to exist in neutron matter and those studied in ultracold
gases, supported by experiments that are relevant for neutron stars.

##  2 Vortex structure and thermodynamic properties

The discovery of He-II triggered studies of quantum vortices, which could be
rather easily generated in this system. However, in superfluid He-II, the
healing length is relatively small, and the vortex core sizes do not exceed a
few Å [31]. Consequently, they cannot be directly observed in experiments. The
situation is different in ultracold atomic gases, which are dilute. Therefore,
the vortex core size can be about $10^{3}-10^{4}$ times larger, depending on
the value of the scattering length. Attempts to cool down fermionic gases
eventually led to the observation of quantum vortices, which provided
unambiguous evidence of superfluidity in the system [32]. The regular lattice
of vortices has been detected in fermionic gases in a wide range of scattering
lengths ranging from Bardeen-Cooper-Schrieffer (BCS) regime through unitary
limit to molecular Bose-Einstein Condensate (BEC) regime. The versatility of
ultracold atomic gases allows for considering properties of vortices within
the whole range of BEC-BCS crossover. It is particularly important as one can
smoothly transform fermionic vortices into bosonic vortices, varying the value
of the scattering length.

Both in the BEC and BCS regimes, the presence of a quantum vortex is
associated with a particular configuration of the order parameter, which
vanishes in the vortex core. The same is valid for superfluid systems, e.g.
He-II, where the "condensate wave function" plays the role of the order
parameter. In both cases, the order parameter is responsible for the stability
and dynamics of quantum vortices. The main difference, however, between
bosonic and fermionic vortices lies in their structure, which, in the latter
case, is more complex. In the bosonic systems at $T=0$ the vortex core is
essentially empty. With increasing temperature, thermal excitations can be
generated, forming an admixture of a normal component inside the core.
Regarding fermionic vortices, the particle density in the core does not vanish
even at $T=0$. Indeed, the order parameter in the vicinity of the center of
the core creates a particular shape of the pairing potential and does not
prevent fermions from occupying the core, leading merely to the emergence of
quantized in-gap states. The first estimates for fermion-bound states in the
core have been given in Ref. [33], and therefore, they are sometimes referred
to as Caroli-De Gennes-Matricon states. It was also conjectured that due to
these states, thermodynamic properties of superconductors would be affected,
namely the specific heat at temperature $k_{B}T<|\Delta|$ ($|\Delta|$ denotes
the magnitude of pairing field far from the vortex) will contain contribution
behaving linearly as a function of $T$. When one goes from a deep BCS regime
to the unitary regime, the spectrum of states in the core gets gradually more
sparse. The in-gap states are expected to disappear eventually beyond the
unitary regime on the BEC side [34].

Before discussing properties of the core states let us first summarize briefly
Bogoliubov-de Gennes (BdG) formalism, which allow to capture the main
properties of the quantum vortex. The explicit form of BdG equations for spin-
imbalanced system reads (no spin-orbit coupling is considered):

$\displaystyle\begin{gathered}{\cal H}\begin{pmatrix}u_{n,\uparrow}(\bm{r})\\\
u_{n,\downarrow}(\bm{r})\\\ v_{n,\uparrow}(\bm{r})\\\
v_{n,\downarrow}(\bm{r})\end{pmatrix}=E_{n}\begin{pmatrix}u_{n,\uparrow}(\bm{r})\\\
u_{n,\downarrow}(\bm{r})\\\ v_{n,\uparrow}(\bm{r})\\\
v_{n,\downarrow}(\bm{r})\end{pmatrix},\\\ {\cal
H}=\begin{pmatrix}h_{\uparrow}(\bm{r})-\mu_{\uparrow}&0&0&\Delta(\bm{r})\\\
0&h_{\downarrow}(\bm{r})-\mu_{\downarrow}&-\Delta(\bm{r})&0\\\
0&-\Delta^{*}(\bm{r})&-h^{*}_{\uparrow}(\bm{r})+\mu_{\uparrow}&0\\\
\Delta^{*}(\bm{r})&0&0&-h^{*}_{\downarrow}(\bm{r})+\mu_{\downarrow}\end{pmatrix},\end{gathered}$
(3)

where $\mu_{\uparrow,\downarrow}$ are chemical potentials for spin-up and
spin-down particles, respectively. Single particle hamiltonian
$h_{\uparrow}=h_{\downarrow}=-\frac{\hbar^{2}}{2m}\nabla^{2}$ consists of the
kinetic term and the mean-field potential, but we will omit the latter, as of
the secondary importance in this case. In principle, the single particle
hamiltonian $h_{\sigma}$ may explicitly depend on the spin state (see eg. Ref.
[35]). The pairing gap is related to quasi-particle wave-functions:

$\Delta(\bm{r})=-\dfrac{g_{\textrm{eff}}}{2}\sum_{0<E_{n}<E_{c}}(u_{n,\uparrow}(\bm{r})v_{n,\downarrow}^{*}(\bm{r})-u_{n,\downarrow}(\bm{r})v_{n,\uparrow}^{*}(\bm{r})),\\\
$ (4)

where $g_{\textrm{eff}}$ is a regularized coupling constant and $E_{c}$ is
cut-off energy scale, see [35] for details of the regularization scheme.

The BdG equations (3) decouple into two independent sets:

$\displaystyle\begin{gathered}\begin{pmatrix}h_{\uparrow}(\bm{r})-\mu&\Delta(\bm{r})\\\
\Delta^{*}(\bm{r})&-h^{*}_{\downarrow}(\bm{r})+\mu\end{pmatrix}\begin{pmatrix}u_{n,\uparrow}(\bm{r})\\\
v_{n,\downarrow}(\bm{r})\end{pmatrix}=E_{n+}\begin{pmatrix}u_{n,\uparrow}(\bm{r})\\\
v_{n,\downarrow}(\bm{r})\end{pmatrix},\end{gathered}$ (6)
$\displaystyle\begin{gathered}\begin{pmatrix}h_{\downarrow}(\bm{r})-\mu&-\Delta(\bm{r})\\\
-\Delta^{*}(\bm{r})&-h^{*}_{\uparrow}(\bm{r})+\mu\end{pmatrix}\begin{pmatrix}u_{n,\downarrow}(\bm{r})\\\
v_{n,\uparrow}(\bm{r})\end{pmatrix}=E_{n-}\begin{pmatrix}u_{n,\downarrow}(\bm{r})\\\
v_{n,\uparrow}(\bm{r})\end{pmatrix},\end{gathered}$ (8)

where $\mu=\frac{1}{2}(\mu_{\uparrow}+\mu_{\downarrow})$ denotes mean chemical
potential and $E_{n\pm}=E_{n}\pm\frac{\Delta\mu}{2}$ with
$\Delta\mu=\mu_{\uparrow}-\mu_{\downarrow}$. Solutions of equations (6) and
(8) are connected through the symmetry relation, namely if vector
$\varphi_{+}=\left(u_{n\uparrow},v_{n\downarrow}\right)^{T}$ represents a
solution of Eq. (6) with eigenvalue $E_{n}$, then vector
$\varphi_{-}=(v_{n\uparrow}^{*},u_{n\downarrow}^{*})^{T}$ is a solution of Eq.
(8) with eigenvalue $-E_{n}$. Therefore it is sufficient to solve equations
(6) only (for all quasiparticle energy states), and then solutions with
positive quasiparticle energies contribute to the spin-down densities, whereas
solutions with negative energies to the spin-up densities.

For the solution describing a vortex the particular form of pairing field is
required $\Delta(\boldsymbol{r})=|\Delta(r)|\exp(i\phi)$ (for the vortex
rotating clockwise), where $(r,\phi,z)$ are cylindical coordinates with $r$
being the distance from the center of the vortex core and $z$ \- the
coordinate along the vortex line (which is assumed to be a straight line).
Therefore the pairing field does not depend on $z$ and the general solution of
eq. (3) describing the vortex reads:

$\displaystyle\begin{gathered}\begin{pmatrix}u_{n,\uparrow}(\bm{r})\\\
v_{n,\downarrow}(\bm{r})\end{pmatrix}=\begin{pmatrix}u_{nmk_{z},\uparrow}(r)e^{im\phi}e^{ik_{z}z}\\\
v_{nmk_{z},\downarrow}(r)e^{i(m-1)\phi}e^{ik_{z}z}\end{pmatrix},\end{gathered}$
(10)

The other pair of solutions can be obtained using the symmetry properties of
BdG equations. Note that (10) implies that the angular momentum of spin-up
particles (holes) differs from spin-down holes (particles), of the same
energy, by $\hbar$.

The spectrum of quantized states inside the vortex core has a peculiar
structure, reflecting the behavior of the order parameter. In the deep BCS
limit, the main features of the spectrum can be reproduced within the
semiclassical Andreev approximation [36]. In this approximation one assumes
the separation of scales related to the density and the pairing field spatial
modulation. The former is set by the Fermi wavelength, whereas the latter - by
the coherence length $\xi$. Therefore one may decompose the variation of $u$
and $v$ components of wave-functions (see Eq. (3)) at the Fermi surface into
rapidly oscillating parts associated with the Fermi momentum $k_{\textrm{F}}$
and smooth variations governed by the coherence length, i.e.
$u(\bm{r})=e^{i\bm{k}_{\textrm{F}}\cdot\bm{r}}\tilde{u}(\bm{r})$ with
$|\bm{k}_{\textrm{F}}|=k_{\textrm{F}}$, and similarly for the $v$ component.
The Andreev approximation can be used also in the case of spin imbalance
systems, providing the local polarization is relatively weak
$\Delta\mu=\mu_{\uparrow}-\mu_{\downarrow}\ll\frac{1}{2}(\mu_{\uparrow}+\mu_{\downarrow})\approx\varepsilon_{F}=k_{\textrm{F}}^{2}/2$.
We will focus only on one set of BdG equations, which in Andreev approximation
describing states close to the Fermi surface acquire the form (we set
$\hbar=m=1$):

$\left(\begin{array}[]{cc}-i\bm{k}_{\textrm{F}}\cdot\bm{\nabla}&\Delta(\textbf{r})\\\
\Delta^{*}(\textbf{r})&i\bm{k}_{\textrm{F}}\cdot\bm{\nabla}\end{array}\right)\left(\begin{array}[]{cc}\tilde{u}_{n,\uparrow}(\textbf{r})\\\
\tilde{v}_{n,\downarrow}(\textbf{r})\end{array}\right)=\tilde{E}_{n+}\left(\begin{array}[]{cc}\tilde{u}_{n,\uparrow}(\textbf{r})\\\
\tilde{v}_{n,\downarrow}(\textbf{r})\end{array}\right),$ (11)

where $\tilde{E}_{n+}=E_{n+}+\frac{\Delta\mu}{2}$. The second pair of
equations for $\tilde{u}_{n,\downarrow}(\textbf{r})$ and
$\tilde{v}_{n,\uparrow}(\textbf{r})$ has similar form and correspond to
$\tilde{E}_{n-}=E_{n-}-\frac{\Delta\mu}{2}$.

Although, Andreev approximation requires that $|\Delta|/\varepsilon_{F}\ll 1$
($\varepsilon_{F}$ denotes the Fermi energy), the spectrum of low-lying states
can be qualitatively reproduced using a schematic model with pairing field
defined as $\Delta(r,\phi)=|\Delta|e^{i\phi}\theta(r-r_{v})$ and shown in Fig.
1(a). The reason is that, the most important ingredient of the model is
related to the particular phase pattern of the pairing potential, whereas the
asymptotic behavior of the wave functions play only a minor role (for a
comprehensive discussion of the quality of Andreev approximation see Ref.
[37]).

Figure 1: A schematic picture of sections through the vortex core. Section
perpendicular to the vortex line is shown in the left subfigure a). The
classical trajectory representing particle of momentum $k_{F}$ is denoted by a
red solid line, and the retroreflected hole is shown as a brown dashed line.
Note that the angular momentum component $L_{z}$ corresponding to the
trajectory is negative, and the vortex rotates counterclockwise. Section along
to the vortex line is shown in the right subfigure b). The classical
trajectory representing particle of momentum $k_{p}$ is denoted by a red solid
line, and the reflected hole of momentum $k_{h}$ is shown as a brown dashed
line.

Then, the quantization condition, originating from the perfect particle-hole
retroreflection, reads [38]:

$\displaystyle\frac{E_{n}}{\varepsilon_{F}}k_{\textrm{F}}r_{v}\sqrt{1-\left(\frac{L_{z}}{k_{\textrm{F}}r_{v}}\right)^{2}}+\arccos{\left(\frac{-L_{z}}{k_{\textrm{F}}r_{v}}\right)}-\arccos{\frac{E_{n}}{|\Delta|}}=\pi
n,$ (12)

where $n\in\\{0,\pm 1,\pm 2,\dots$}, $r_{v}$ denotes the radius of the vortex
core, $k_{\textrm{F}}$ \- Fermi momentum, and $|L_{z}|=rk_{\textrm{F}}$. Note
that only the states with $n=0$ correspond to core states, i.e.
$E_{n=0}\lesssim|\Delta|$111While solving the full BdG eqs. the states
corresponding to $n\neq 0$ can still be found at energies smaller than
$|\Delta|$, but they are very close to continuum.. The limit $|E|\ll|\Delta|$
can be quite accurately approximated by the expression:

$E_{\pm,n=0,m}\approx-\frac{|\Delta|^{2}}{\varepsilon_{F}\frac{r_{v}}{\xi}\left(\frac{r_{v}}{\xi}+1\right)}m,$
(13)

where $m$ is the magnetic quantum number associated with $L_{z}=\hbar m$,
pointing along the vortex axis and
$\xi=\varepsilon_{F}/(k_{\textrm{F}}|\Delta|)$ is a coherence length. Minus
sign in front of the rhs expression (13) is related to the counterclockwise
rotation of the vortex. Linear dependence $E\propto L_{z}$ holds until the
energy approaches $\Delta$, and subsequently, it is bent towards large angular
momenta. This part of the spectrum cannot be reproduced within Andreev
approximation. In the regime close to unitarity and the inner crust of neutron
stars, the linear part of the spectrum consists of a few states only, as shown
in the inset of Fig. 2. The results presented in this figure have been
obtained by solving numerically Hartree-Fock-Bogoliubov equations (25) (having
the same form as eqs. (3)) on the spatial lattice (in 3D) for the pure neutron
matter. The BSk energy density functional with the local pairing field
$\Delta(\boldsymbol{r})$ has been used (see Section 3). The resulting profile
of the pairing field and the neutron density distribution
$\rho(\boldsymbol{r})$, have been obtained selfconsistently through the total
energy minimization, with the requirement that the pairing field possess the
structure characteristic for the vortex located in the center of the
cylindrical box. Thus the results represent more realistic structure of the
vortex core. The quasiparticle energies as a function of the angular momentum
along the vortex symmetry axis have been plotted in the inset. The plotted
energies correspond to $k_{z}=0$ in expression (10).

Figure 2: The cross-section through the center of the vortex in neutron matter
of bulk density $\rho_{\infty}=0.0059$ fm-3 and corresponding bulk pairing gap
$\Delta_{\infty}=1.33$ MeV. The parameters correspond to the inner crust
region of a neutron star. The cross-section of (a) density and pairing field
(b) (both quantities normalized to the corresponding bulk value). Two
characteristic length scales are shown: inverse Fermi momentum $k_{F}^{-1}$
and the superfluid coherence length $\xi$. Inset in (b): the energy spectrum
of quantum states (of particle and hole types) and their quantized angular
momenta (the relative shift of angular momenta between particles and holes is
the consequence of relation (10)). The energy scales governing the physics of
the system — bulk energy gap $\Delta_{\infty}$, as well as so-called minigap
$E_{mg}$ — are shown. The results are obtained with DFT framework for nuclear
systems [28].

Note that in the inner crust of the neutron star, in the region where the
s-wave pairing is predicted to be the strongest, i.e. with the value of the
pairing gap within the interval $(1.2,1.6)$ MeV, the size of the vortex core
is much smaller (at $T\lesssim 0.1$ MeV) than the typical modulation of the
nuclear density, induced by the presence of nuclear lattice. Namely, the radii
of the vortex cores do not exceed $12$ fm [28], whereas typical lattice
constants in this region range between $40$ and $70$ fm [39]. Therefore, for
the nuclear densities $(0.006-0.034)$ fm-3 it is possible to place the vortex
core in the region, where the matter is uniform. The situation is different,
however, if one approaches the boundaries of the inner crust. In the limit of
small densities, at the border of the outer crust region, the lattice constant
reaches values of about $100$ fm, whereas the size of the vortex cores can be
arbitrarily large due to the vanishing density of the neutron gas outside
nuclei. The similar situation occurs in the region, where pasta phases are
predicted to exist, i.e. at the densities $0.05$ fm-3 and higher. The typical
length scale of nuclear density modulation is about $20$ fm and the vortex
core can be of similar size. In these cases one has to take into account that
the density inside the vortex core is in addition modulated due to the
particular nuclear structure, independent of the shape of the pairing field
associated with the vortex. This effect will in turn introduce additional
modification of the structure of the in-gap states, due to the scattering of
quasiparticles on nuclear inhomogeneities. Moreover, in these region, the
scattering of quasiparticles (inside the vortex core) on nuclear
inhomogeneities may provide another efficient source of energy dissipation for
a moving vortex (see discussion in Section 3).

There are few immediate consequences to the existence of the so-called chiral
branch corresponding to $n=0$. Since the quasiparticle excitation spectrum
consists of states with $m<0$ (for a vortex rotating counterclockwise), the
energy of the first excited state
$E_{mg}\approx\frac{|\Delta|^{2}}{2\varepsilon_{F}}$ (assuming $r_{v}=\xi$)
sets the energy scale (minigap energy) associated with the excitation of the
vortex core. Moreover, the chiral branch’s structure determines the density
behavior in the vortex core. Specifically, since the $m=0$ state is
unoccupied, the density reveals depletion in the core. It can be deduced from
BdG eqs. that density $\rho(r)\propto r^{2}+\textrm{const}$ in the vicinity of
the center of the vortex. The vortex core structure, obtained within the
framework of Density Functional Theory (see section 3), is shown in Fig. 2. It
has to be emphasized that the depletion of the density inside a vortex core is
crucial for the experimental detection of vortices in ultracold Fermi gases.

The presence of a chiral branch has an impact on thermodynamic properties as
well. For temperatures within the range $E_{mg}\ll k_{B}T\ll|\Delta|$, it was
predicted [33] that the specific heat should be a linear function of
temperature. It is, however, true in deep BCS limit only
($\Delta\ll\varepsilon_{F}$). Due to the limited number of in-gap states in
strongly interacting systems, the temperature range fulfilling the above
criterion may not exist. Moreover, the dependence of the size of the core on
temperature makes the functional dependence of the specific heat more
complicated [28]. Another modification of the specific heat comes from the
superflow surrounding a vortex due to the rearrangement of the quasiparticle
spectrum induced by the flow. Namely, the quasiparticle dispersion relation in
the presence of superflow with velocity $v_{s}$ reads:

$E_{\pm}(\boldsymbol{k},\boldsymbol{v_{s}})=\hbar\boldsymbol{k}\cdot\boldsymbol{v_{s}}\pm\sqrt{\left(\frac{\hbar^{2}k^{2}}{2M}-\tilde{\mu}\right)^{2}+|\Delta|^{2}},$
(14)

where $\tilde{\mu}=\mu-\frac{1}{2}Mv_{s}^{2}$ represents the correction to the
chemical potential due to the superflow. We consider the case $\hbar
k_{\textrm{F}}v_{s}/|\Delta|\ll 1$ corresponding to velocities smaller than
the critical velocity. In that case
$E_{+}(\boldsymbol{k},\boldsymbol{v_{s}})>0$ and one can determine the
correction to the specific heat for the moving superfluid. Keeping terms up to
$v_{s}^{2}$ one gets:

$\displaystyle C_{V}(v_{s})\approx\frac{2}{T^{2}}VN(0)\sqrt{2\pi
T|\Delta|}\exp\left(-\frac{|\Delta|}{T}\right)$
$\displaystyle\times\left[|\Delta|^{2}+\frac{\mu
Mv_{s}^{2}}{3}\left(\left(\frac{|\Delta|}{T}\right)^{2}-4\frac{|\Delta|}{T}+2-\frac{3}{4}\left(\frac{|\Delta|}{\mu}\right)^{2}\right)^{2}\right],$
(15)

where

$N(0)=\frac{1}{(2\pi)^{2}}\left(\frac{2M}{\hbar^{2}}\right)^{3/2}\sqrt{\mu}$
(16)

is the density of states at the Fermi surface. The last term, proportional to
$\left(|\Delta|/\mu\right)^{2}$, is a correction due to the modification of
the chemical potential $\tilde{\mu}$ in the presence of superflow. It is,
however, at least an order of magnitude smaller than the other terms and can
be neglected. One may apply this formula to the flow induced by a vortex
within the local density approximation. Namely, one can divide the volume
around the vortex line into infinitesimal concentric cylindrical shells (of
length $L$), in which the superfluid velocity is constant, and subsequently
integrate the contributions to the specific heat coming from each shell. For
the cylinder of radius $R_{out}$ one gets the correction to specific heat per
unit length of the vortex:

$\displaystyle\frac{\Delta C_{V}^{flow}}{L}\approx$
$\displaystyle\frac{1}{3}(2\pi)^{3/2}N(0)\sqrt{\frac{|\Delta|}{T}}\frac{\mu}{T}\frac{\hbar^{2}}{2M}\ln\left(\frac{R_{out}}{r_{v}}\right)\left[\left(\frac{|\Delta|}{T}\right)^{2}-4\frac{|\Delta|}{T}+2\right]\exp\left(-\frac{|\Delta|}{T}\right).$
(17)

As discussed in Ref. [28], $R_{out}$ is of the order of inter-vortex distance
in the case of vortex lattice.

Consequently, the modification of the specific heat per unit volume, due to
the presence of vortices, reads:

$\frac{1}{V}\Delta C_{V}=\sigma_{vor}\left(\frac{\Delta
C_{V}^{core}}{L}+\frac{\Delta C_{V}^{flow}}{L}\right),$ (18)

where $\sigma_{vor}=N_{vor}/S$ (and $V=S\times L$) is the surface vortex
density and the two terms: $\Delta C_{V}^{core}$, $\Delta C_{V}^{flow}$
represent modification of the specific heat due to the vortex core and the
flow induced by the vortex, respectively. Due to small average vortex density
$\sigma_{vor}\approx 10^{-21}$ fm-2, the modification of the specific heat of
the neutron star crust is negligible for the temperatures $T\gtrapprox 0.1$
MeV. However, if the vortex lattice is perturbed, giving rise to the
turbulent-like behavior, one can expect that the vortex density will exhibit
significant spatial fluctuations [40]. Using the results of Ref. [28] (see
Fig. 10 therein) one can estimate that the local increase of vortex density to
$\sigma_{vor}\approx 10^{-2}$ fm-2 leads to approximately twice larger
specific heat at $T\approx 0.1$ MeV for a wide range of neutron densities in
the crust $(0.006-0.034)$ fm-3. For even lower $T$, specific heat will
increase exponentially.

The core states, discussed above, correspond to 2D vortex, i.e., they are
characterized by $k_{z}=0$, where $k_{z}$ is a momentum along the vortex line.
In 3D, corresponding to the infinite vortex line, each state forming the
chiral branch represents a band. The slope of the band is directly related to
the effective mass of quasiparticle excitations, and it can be traced back to
the quasiparticle scattering properties along the vortex line. In Fig. 1(b),
the schematic picture of Andreev scattering, leading to the motion of
quasiparticles along the vortex line, has been shown. Contrary to the
quantization condition, which resulted from the assumption that the
hole(particle) is reflected exactly backward (which is true if the incoming
particle(hole) is at the Fermi surface), here we need to release this
constraint to determine the effective mass. Consequently one needs to take
into account corrections related to $\varepsilon_{F}\pm E$. As a result of the
momentum conservation along the vortex line, the reflection law reads:
$\sqrt{\varepsilon_{F}+E}\sin\alpha=\sqrt{\varepsilon_{F}-E}\sin\beta$, where
$k_{p}=\sqrt{2(\varepsilon_{F}+E)}$ and $k_{h}=\sqrt{2(\varepsilon_{F}-E)}$
are particle and hole momenta, respectively. The effective velocity along the
vortex line can be evaluated as $v_{z}=\frac{S}{\Delta
T}=\sqrt{2(\varepsilon_{F}+E)}\sin\alpha\sin(\beta-\alpha)/\sin(\beta+\alpha)$,
where $\Delta T$ denotes the time interval between two consecutive particle-
to-hole reflections. Finally one gets [38]:

$v_{z}=k_{z}\frac{\sqrt{k_{p}^{2}-k_{z}^{2}}-\sqrt{k_{h}^{2}-k_{z}^{2}}}{\sqrt{k_{p}^{2}-k_{z}^{2}}+\sqrt{k_{h}^{2}-k_{z}^{2}}},$
(19)

where $k_{z}=k_{p}\sin\alpha=k_{h}\sin\beta$ is the momentum component along
the vortex line. Considering the linear term in $k_{z}$ and $E$ on the rhs,
one obtains the effective mass $M_{\textrm{eff}}^{-1}\approx
E/2\varepsilon_{F}$, which agrees with the effective mass derived from the
formula for the dispersion relation in the BCS limit
$E(k_{z})=E(0)/\sqrt{1-k_{z}^{2}/(2\varepsilon_{F})}$ [33]. One may also
estimate the magnitude of the effective mass component along the vortex line,
corresponding to angular momentum $L_{z}=\hbar m$:
$M_{\textrm{eff}}^{-1}(m)\approx\frac{2|m|}{3}\left(\frac{\Delta}{\varepsilon_{F}}\right)^{2}$.
Note that, in deep BCS limit, the inverse of the effective mass will be
exponentially small since $\Delta/\varepsilon_{F}\propto
e^{-\pi/2|a|k_{\textrm{F}}}$, and clearly, the departure from the flat band
behavior will be significant at the unitarity, where
$\Delta/\varepsilon_{F}\approx 0.5$. The band flatness, increasing the
effective mass, will affect the propagation of the confined quasiparticle
excitations, e.g., in the form of local spin-polarization, along the vortex
line. It was shown in Ref. [41] that the locally induced spin-polarization in
the vortex core (e.g., due to the reconnection process with spin-polarized
vortex) can hardly propagate along the vortex line. Indeed, the motion along
the vortex line is characterized by the velocity
$v_{z}=k_{0}/M_{\textrm{eff}}\propto
k_{0}\left(\frac{\Delta}{\varepsilon_{F}}\right)^{2}$, where $k_{0}$ is the
initial momentum of the wave packet. Similarly, the wave packet width, in the
limit of long times, behaves as $\sqrt{\langle(z-v_{z}t)^{2}\rangle}\propto
t\left(\frac{\Delta}{\varepsilon_{F}}\right)^{2}$ and leads to an effective
suppression of the polarization propagation, see Ref. [38].

The structure of the vortex core is sensitive to spin imbalance. When the
chemical potentials for spin-up ($\mu_{\uparrow}$) and spin-down
($\mu_{\downarrow}$) fermions begin to differ, the core of the vortex will be
affected first, before the superfluid will be modified in bulk. This is due to
the fact that $E_{mg}<|\Delta|$. Indeed, unpaired fermions tend to accumulate
at the core, as shown in [42, 43]. Consequently, the chiral branch splits into
two components corresponding to spin-up and spin-down particles. The main
features of this splitting can be described introducing a slight modification
of the expression (12), which reads:

$\displaystyle\frac{\tilde{E}_{n\pm}}{\varepsilon_{F}}k_{\textrm{F}}r_{v}\sqrt{1-\left(\frac{L_{z}}{k_{\textrm{F}}r_{v}}\right)^{2}}+\arccos{\left(\frac{-L_{z}}{k_{\textrm{F}}r_{v}}\right)}-\arccos{\frac{\tilde{E}_{n\pm}}{|\Delta|}}=\pi
n,$ (20)

where $\tilde{E}_{n\pm}=E_{n\pm}\pm\frac{\Delta\mu}{2}$ and
$\Delta\mu=\mu_{\uparrow}-\mu_{\downarrow}$ is the difference between chemical
potentials of spin-up and spin-down particles, respectively. A schematic
representation of the solutions corresponding to the chiral branch ($n=0$) is
shown in Fig. 3.

Figure 3: Chiral branches (red solid line) of low-lying in-gap states in the
vortex core for clockwise rotating vortex. Parts of branches that became empty
and occupied due to induced spin imbalance have been shown in green and brown
colors, respectively.

The immediate consequence of this splitting results in occupying states, by
majority spin particles, which rotate in opposite directions than the vortex.
The highest value of angular momentum corresponding to the opposite rotation
$L_{z}^{max}=\hbar m_{opposite}$ is related to spin imbalance and reads:

$\max|m_{\textrm{opposite}}|\approx\frac{1}{2}\frac{\varepsilon_{F}}{|\Delta|^{2}}\frac{r_{v}}{\xi}\left(\frac{r_{v}}{\xi}+1\right)\Delta\mu.$
(21)

It is, therefore, clear that the reversed flow of these particles partly
cancels the flow of the unpolarized component. However, as was shown in
Ref.[38], the cancellation is not exact and leads to a reversal of the total
flow in the core. The splitting of the chiral branch, induced by the spin
imbalance, which rearranges the occupation of low-lying states, has another
effect. Namely, the $m=0$ state becomes now occupied, and the density is no
longer depleted inside a polarized vortex. It makes the experimental detection
of vortices in spin imbalanced system far more complex.

In the case of quantum vortices in the inner crust of a neutron star, the
mechanism that may lead to the polarization of the vortex core is due to the
magnetic field. It couples to the neutron magnetic moment and may induce a
spin flip. In Ref. [28] the minimal value of the magnetic field required to
produce such an excitation in the vortex core has been estimated to range
between about $10^{15}-10^{16}G$, depending on neutron density. Note, that
this value is more than an order of magnitude smaller than magnetic field
needed to destroy superfluidity in the neutron star crust, which was estimated
as $B\geq 10^{17}G$ [44]. There are known magnetars where such conditions may
exist [45].

The important effect of polarization in the vortex core is that it eventually
causes the minigap to vanish. As was shown in Ref. [38], this is due to the
fact that the spectrum of states with $k_{z}=0$ is asymmetric with respect to
Fermi surface (see Fig. 3). However, the spectrum of states corresponding to
$k_{z}\gg k_{\textrm{F}}$ becomes symmetric, which is apparent by inspecting
the structure of the BdG Hamiltonian, describing the infinite vortex line.
Therefore, one may infer that at certain values of $k_{z}=\pm k_{z1},\pm
k_{z2},\dots$ the quasiparticle energies vanishes $E(\pm k_{zi})=0$. When the
quasiparticle energy changes from negative to positive value, the particle
state $v_{\uparrow}$ with momentum $m$ is converted into the hole state
$u_{\uparrow}$ with momentum $-m+1$, inducing the change of $m$: $\Delta
m=|2m-1|$. Therefore, the spin-imbalanced vortex is characterized by a series
of quasiparticle level crossings at the Fermi surface, i.e., points at which
the minigap vanishes. Eventually, if the spin imbalance increases, it is
predicted that modulation of the pairing field, similar to the Fulde-Ferrell-
Larkin-Ovchinnikov (FFLO) phase, will be seen [46].

All these characteristics of the fermionic vortex core structure should have
an impact on dynamics and, in particular, on dissipative processes associated
with the vortex motion. These will be discussed in the next section.

##  3 Vortex dynamics

The problem of vortex dynamics is essential for understanding the behavior of
superfluids, particularly the role of dissipative processes in the decay of
quantum turbulence [47, 48]. The apparent differences between the structure of
bosonic and fermionic vortices, discussed in the previous section, raise a
question about the impact on vortex dynamic. The existence of another energy
scale associated with minigap and the presence of the chiral branch is
expected to generate additional contributions to the dissipative force. Last
but not least, there is still an open question associated with vortex inertia,
which in the case of bosonic vortices is usually assumed to be negligible
[49].

Answers to these questions should result in formulating the effective equation
of motion of a fermionic vortex, where all terms will acquire microscopic
underpinning. If we consider a 2D vortex, i.e., neglecting degrees of freedom
associated with the susceptibility of vortex lines to bending and generating
Kelvin waves, the problem is still complex as the most general equation of
motion comprises the following terms:

$M_{V}\frac{d^{2}\boldsymbol{R}_{V}}{dt^{2}}=\gamma\frac{d\boldsymbol{R}_{V}}{dt}+\omega\frac{d\boldsymbol{R}_{V}}{dt}\times\boldsymbol{e}_{z}+\boldsymbol{F}_{\textrm{superfluid}}+\boldsymbol{F}_{\textrm{pinning}},$
(22)

where $\boldsymbol{R}_{V}$ is the vortex position, $\boldsymbol{e}_{z}$ is the
unit vector perpendicular to the plane of vortex motion,
$\boldsymbol{F}_{\textrm{pinning}}$ describes the interaction with
inhomogeneities and $\boldsymbol{F}_{\textrm{superfluid}}$ is the component of
the force acting on a vortex but independent of its velocity. The equation
arises from the general expression of forces that can be present in the system
and being in agreement with Galilean invariance. The latter requirement
implies that the force either depends on
$(\frac{d\boldsymbol{R}_{V}}{dt}-\boldsymbol{v}_{s})$ or on
$(\frac{d\boldsymbol{R}_{V}}{dt}-\boldsymbol{v}_{n})$, where
$\boldsymbol{v}_{s}$ and $\boldsymbol{v}_{n}$ describe the velocity of
superfluid and normal components, respectively. Consequently, the total force
on rhs of (22) reads:

$\displaystyle\boldsymbol{F}$ $\displaystyle=$ $\displaystyle
A\boldsymbol{e}_{z}\times\left(\frac{d\boldsymbol{R}_{V}}{dt}-\boldsymbol{v}_{s}\right)+B\boldsymbol{e}_{z}\times\left(\frac{d\boldsymbol{R}_{V}}{dt}-\boldsymbol{v}_{n}\right)+C\left(\frac{d\boldsymbol{R}_{V}}{dt}-\boldsymbol{v}_{n}\right)+$
(23) $\displaystyle+$
$\displaystyle\boldsymbol{F}_{\textrm{pinning}}\left(\boldsymbol{R}_{V},\frac{d\boldsymbol{R}_{V}}{dt},\boldsymbol{e}_{z}\times\frac{d\boldsymbol{R}_{V}}{dt}\right),$

where the last term, describing the pinning force, breaks Galilean invariance
as it corresponds to the interaction with impurities222More precisely: the
pinning force describes all effects related to the momentum transfer between
the vortex and impurities. It may depend on the location of the vortex, as
well as its velocity [50]. The dependence on the vortex velocity becomes
important in the limit of large core sizes (as compared to the impurity size),
and originates from the effect of scattering of quasiparticles, bound in the
vortex core. It gives rise to the so-called Kopnin-Kravtsov force [51, 52]. In
the case of the neutron star crust this effect may become important in the
region of pasta phases, where the core size becomes larger (due to weaker
pairing correlations) than the typical scale of spatial density modulation.
The pinning force is particularly important in the context of neutron star
crust [53, 54], where the pinning-unpinning mechanism is essential for
understanding the glitch phenomenon [26, 48]. The coefficients $A,B,C$ depends
on densities $\rho_{s}$ and $\rho_{n}$, describing superfluid and normal
components of the system. The equation (22) is obtained by separating terms
proportional to $\frac{d\boldsymbol{R}_{V}}{dt}$, and introducing
$\boldsymbol{F}_{\textrm{superfluid}}$, which depends on $\boldsymbol{v}_{s}$
and $\boldsymbol{v}_{n}$ only (see eg. Ref. [55]). For the sake of generality
we include the vortex mass $M_{V}$, which, however, is routinely assumed to be
negligible. Coefficients $\gamma$ and $\omega$ need to be derived from
microscopic theory and are functions of densities $\rho_{s}$ and $\rho_{n}$.

There are still many uncertainties concerning each term of this equation,
particularly for fermionic vortices. Apart from the Magnus force
$\boldsymbol{F}_{M}\propto\rho_{s}\boldsymbol{e}_{z}\times(\frac{d\boldsymbol{R}_{V}}{dt}-\boldsymbol{v}_{s})$,
which has similar form in both bosonic and fermionic case, the form and
magnitude of dissipative forces, both transversal and longitudinal, are still
debatable [56, 57, 58]. In the microscopic description, these forces have to
be related to quasiparticles and phonon scattering on the vortex core. It
includes the so-called Iordanskii force, which is the transverse force related
to the relative motion with respect to the normal component of the superfluid:
$\boldsymbol{F}_{I}\propto\boldsymbol{e}_{z}\times\left(\frac{d\boldsymbol{R}_{V}}{dt}-\boldsymbol{v}_{n}\right)$.
It is predicted to emerge from the scattering of elementary excitations of
superfluid on the vortex core [59]. In the BEC case, it is related to the
scattering of phonons, whereas in Fermi systems it would include also
quasiparticle excitations. However the role and the magnitude of this force
are still not clear (see eg. the argument in Ref. [60] against the existence
of the transverse force, associated with the normal component).

Usually, to date, the determination of dissipative forces relied on a
semiclassical approach [61, 62]. It consisted of modeling the presence of the
classical Hamilton function: $H(L_{z})=\alpha L_{z}$ grasping the main feature
of the chiral branch. The evolution of quasiparticles has been modeled through
the semiclassical distribution function $f(L_{z},\phi)$ , where $\phi$ is an
angle being canonically conjugate variable to $L_{z}$. Subsequently, one may
search for a solution of Boltzmann eq. describing small deviations from
equilibrium distribution. The quasiparticle collisions have been treated in
relaxation time approximation, which introduces another parameter to the
description. This approach is, however, expected to work in the deep BCS
regime, where one can ignore the presence of discrete states in the band, and
the effects related to mini-gap are of secondary importance.

Recently, a microscopic approach rooted in density functional theory has
become possible. It provides a framework for comprehensive studies of static
and time-dependent problems including thermal effects. Its popularity in
nuclear physics grows with time, and today it is used for describing the
properties of nuclei across the whole nuclear chart, nuclear reactions, and
properties of neutron star crust [63, 64, 65, 66]. The modern DFT variants
takes into account superfluid correlations, with the pairing field treated as
a dynamical variable. One of the variants that proved its usability,
especially in the context of ultracold Fermi gases, is known as Superfluid
Local Density Approximation (SLDA) [35]. The name refers to the class of the
energy density functionals $\mathcal{E}$ that depend only on local densities,
such that the energy of system can be written as

$E=\int\mathcal{E}[\rho_{q}(\bm{r}),\nu_{q}(\bm{r}),\tau_{q}(\bm{r}),\bm{j}_{q}(\bm{r}),\ldots]\,d\bm{r}.$
(24)

In this expression, $q$ is a generalized index that in the context of nuclear
systems refers to protons and neutrons ($q=n,p$), while in the context of cold
fermionic gases - to atomic species. The functional is expressed through
various densities, where the most common are: normal density ($\rho_{q}$),
kinetic density ($\tau_{q}$), current density ($\bm{j}_{q}$) and anomalous
density ($\nu_{q}$). Other densities, not listed here, are indicated by dots.
The anomalous density is crucial for the description of superfluidity as it
quantifies the presence of Cooper pairs and defines the order parameter
$\Delta_{q}(\bm{r})=g\nu_{q}(\bm{r})$ where $g$ is coupling constant. The
order parameter, within this framework, is treated as a complex field,
allowing for the description of quantum vortices.

The precise form of the functional depends on the considered system; however,
for all local functionals that belong to the SLDA class, the energy
minimization condition leads to equations that formally have the same
structure as Hartree-Fock-Bogoliubov or Bogoliubov-de Gennes equations. For
static problems, they have a generic form (spin indices are omitted):

$\displaystyle\begin{pmatrix}h_{q}(\bm{r})&\Delta_{q}(\bm{r})\\\
\Delta^{*}_{q}(\bm{r})&-h^{*}_{q}(\bm{r})\end{pmatrix}\begin{pmatrix}u_{q,n}(\bm{r})\\\
v_{q,n}(\bm{r})\end{pmatrix}=E_{q,n}\begin{pmatrix}u_{q,n}(\bm{r})\\\
v_{q,n}(\bm{r})\end{pmatrix},$ (25)

where $[u_{q,n},u_{q,n}]^{T}$ are quasiparticle orbitals expressed as mixtures
of particles ($v_{q,n}$) and holes ($u_{q,n}$). One has to remember that both
$h_{q}(\bm{r})$ and $\Delta_{q}(\bm{r})$ are in fact $2\times 2$ matrix in
spin space and therefore $[u_{q,n},u_{q,n}]^{T}$ is in fact four component
vector (see eqs. (3)). They are used to construct densities, e.g.,
$\rho_{q}(\bm{r})=\sum_{E_{q,n}>0}|v_{q,n}(\bm{r})|^{2}$. The single-particle
mean-fields $h_{q}$ and pairing potentials $\Delta_{q}$, are defined via
functional derivatives over densities:

$\displaystyle h_{q}$
$\displaystyle=-\bm{\nabla}\frac{\delta\mathcal{E}}{\delta\tau_{q}}\bm{\nabla}+\frac{\delta\mathcal{E}}{\delta\rho_{q}}-\frac{i}{2}\left\\{\frac{\delta\mathcal{E}}{\delta\bm{j}_{q}},\bm{\nabla}\right\\},$
(26) $\displaystyle\Delta_{q}$
$\displaystyle=-\frac{\delta\mathcal{E}}{\delta\nu^{*}_{q}},$ (27)

where $\left\\{\ldots\right\\}$ denotes the anticommutator and
$\frac{\delta\mathcal{E}}{\delta\bm{j}_{q}}$ represents a vector constructed
by variations over three components of the current $\bm{j}_{q}$. One may
notice that expressions, like kinetic term $\frac{1}{2m}\nabla^{2}$, or mean-
field potential $U$ are replaced by generalized forms, and by adequately
choosing the form of the functional one can go beyond the mean-field approach,
while keeping the complexity of the framework at the same level as the mean-
field formulation. This flexibility can be used to create the functional,
which reproduces selected observables, derived by other methods or taken from
experiments. In the case of ultracold atoms the SLDA functional generates the
correct equation of state (and related thermodynamic properties), strength of
the pairing correlations, and effective mass of quasiparticles [67, 35, 68].
The functionals for nuclear systems are more complex, and they are constructed
usually in such a way to reproduce nuclear masses and radii and/or equation of
state of neutron matter or symmetric nuclear matter. For example, the results
presented in Fig. 2 were obtained with Brussels-Montreal Skyrme functional
(BSk), which was specifically constructed for astrophysical applications [69,
70, 71, 72, 73]. It assures correct reproduction of the equation of state and
have been fitted to reproduce the magnitude of the pairing gap as a function
of neutron density, obtained within Brueckner theory, including screening
effects [74]. Consequently, DFT provides a reliable source of microscopic
information about the vortices [34, 75, 76, 77, 28]. The approach can be
extended to time-dependent phenomena by replacing $E_{q,k}\rightarrow
i\hbar\partial/\partial t$, which allows studying the vortex dynamics [41, 27,
78, 79, 80, 81]. Consequently, the time-dependent studies can provide
microscopic insight into origin of forces acting on the vortex.

It has been conjectured that an accelerating vortex may be subject to
dissipative forces originating from quasiparticles trapped in the core [62].
The conjecture, based on a semiclassical approach, presents a mechanism in
which the acceleration of a vortex will lead to heating up a gas of
quasiparticles in the core. As a result, some of them will be emitted from the
core, taking away energy carried by the vortex line. Although the
semiclassical approach is applicable in deep BCS regime, the experiment has
been recently performed for a system close to the unitarity, where significant
dissipation effects have been observed [82, 83]. However, the presence of
finite temperature did not allow to distinguish the source of dissipation
clearly. Through time-dependent SLDA (TDSLDA) simulations of collisions of two
pairs of vortex-antivortex dipoles, it has been possible to identify
dissipative mechanism due to quasiparticle ejection [78]. The application of
the TDSLDA framework revealed that the dissipative mechanism via excitations
of the vortex core, while present, emerges to be of secondary importance, and
thermal effects dominate the dynamics in experimental realization.

It has to be noted that, in the context of neutron stars, there is an
additional dissipation channel associated with neutron quasiparticles bound in
the vortex core. It is due to the interaction both with electrons and protons
and has been discussed in Ref. [84]. In the case of electrons the coupling is
provided by interaction with the neutron magnetic moment, whereas protons
couple strongly through the nuclear interaction. Detailed dynamics is
complicated due to the coupling between electron dynamics and protons [85].
However, the results indicate that coupling of proton-electron plasma to
neutron vortices lead to short relaxation times implying that magnetars
superfluid cores will couple to the remaining stellar plasma on short
dynamical time-scales once proton superconductivity is suppressed.

Figure 4:  Vortex dipole (vortex-antivortex pair) propagating in a spin
imbalanced unitary Fermi gas (UFG). Spin imbalance induces the existence of
nodal lines of the pairing field, where the majority spin particles are
accumulated. Therefore the local polarization is increased along the nodal
lines, which is seen as white stripes (local polarization is defined as:
$p(r)=\frac{\rho_{\uparrow}-\rho_{\downarrow}}{\rho_{\uparrow}+\rho_{\downarrow}}$).
Arrows indicate the local current and shows the position of the vortex cores.
Lower subfigure b) show the configuration after time interval
$t=250\epsilon_{F}^{-1}$ with respect to initial configuration a).

Another successful application of TDSLDA approach has been the determination
of the pinning force acting between a vortex and a nucleus immersed in
superfluid neutron matter. To date, studies of this effect were based on
comparing the energy of the vortex pinned and unpinned from the impurity [86,
87, 88, 89, 75, 90, 91, 92]. However, these results were not very accurate due
to small energy difference between configurations. One must also remember that
the pinning process requires energy dissipation. Otherwise, the vortex will
simply orbit the nucleus as a result of the Magnus force. Therefore, the
proper description of dissipative processes is essential. Ref. [27] described
how the pinning force can be accurately determined by dragging the impurity
against the vortex with constant velocity. Then, by combining the information
about the force with the vortex-nucleus separation, one could extract the
force as a function of vortex-nucleus separation. It was found that the
extracted force has a negligible tangential component and is of predominantly
central character. The effective range of the force is about $10\textrm{fm}$
for the lower density, increasing to about $15\textrm{fm}$ for the higher
density, consistent with an increasing coherence length with density and
decreasing neutron pairing gap. The behavior of the total force for small
separations demonstrates that it is not merely a function of distance. At
small separations, the deformation of the vortex line and the nuclear
deformation become important degrees of freedom.

##  4 Summary and open questions

The vortex structure and dynamics discussed in the previous sections are
crucial for understanding the role of vortices in neutron stars, particularly
in the glitch phenomenon, and more generally, to understand the decay pattern
of fermionic quantum turbulence.

Vortex structure is relatively well known, although its modifications due to
spin imbalance are still not fully understood. The situation is different when
one considers dynamical properties. To date, most investigations were
performed based on very simplified assumptions, which at most can be valid in
deep BCS regimes. However, the possibility of using the unconstrained time-
dependent BdG approach, and more generally, the TDSLDA framework, paved the
way to extract components of forces exerted on a vortex moving through a
superfluid. It whets our appetite concerning the determination of the
microscopic underpinning of vortex dynamics in a fermionic environment. It is
particularly important that these studies are performed in close collaboration
with experiments in ultracold atoms, which allows the testing of the
applicability of the theoretical framework. To date, the two aforementioned
examples of applications of TDSLDA have provided very promising outcomes.

The degree of freedom related to spin imbalance will open a new avenue for
studying vortex dynamics. The motion of the vortex in the presence of a large
number of majority spin particles, may change the vortex dynamics
qualitatively. Indeed, their scattering off the core is expected to enhance
dissipative processes considerably [93]. As an example, in the Fig. 4, the
results of TDSLDA simulations of moving vortex dipole have been shown. In the
lower subfigure, one can notice the significant distortion of the initial
vortex dipole (upper subfigure), which occurs due to interaction with majority
spin particles accumulated in the pairing nodal lines. As a result some
quasiparticles have been trapped in the vortex core. Therefore, it is expected
that the motion of the vortex in the spin-imbalanced system may shed light on
the magnitude of the Iordanskii force, which should be significantly enhanced
in this case. As a result, we may encounter new, unexpected features of vortex
dynamics that have not yet been observed.

Acknowledgments

This work was financially supported by the (Polish) National Science Center
Grants under Contracts No. UMO-2021/43/B/ST2/01191 (PM, AM),
UMO-2022/45/B/ST2/00358 (GW, ABa) and UMO-2021/40/C/ST2/00072 (DP). This work
used computational resources of Tsubame3.0 supercomputer provided by Tokyo
Institute of Technology through the HPCI System Research Project (Project ID:
hp230081).

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