# Notes on the Riemann zeta function

Tanfer Tanriverdi
Harran University Faculty of Arts and Sciences
Department of Mathematics Şanlıurfa Turkey 63290
e-mail<EMAIL_ADDRESS>

This article is dedicated my entire family and people I love

###### Abstract

In this article, some new representations of the Riemann zeta function and
their associated zeros are explored for sufficiently large $n$.

##  1 Introduction

The importance of Riemann zeta function is trivial to anybody who runs into it
while working out with certain subjects such as number theory, complex
analysis, etc. Notes on the Riemann zeta function and its applications, see
[1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. In [7], the Riemann zeta function
$\zeta(z)$ is defined by

$\displaystyle\zeta(z)=\sum_{n=1}^{\infty}\frac{1}{n^{z}},\
\operatorname{Re}(z)>1.$ (1)

The series (1) is convergent and analytic function for
$\operatorname{Re}(z)>1$. If $z=s>1$ is real number then Riemann zeta function
is represented and studied by Euler as

$\displaystyle\zeta(z)=\prod_{p}\left(1-\frac{1}{p^{s}}\right)^{-1}$

where $p$ runs through all prime numbers. There are several other
representations

$\displaystyle\zeta(z)=\frac{1}{\Gamma(z)}\int_{0}^{\infty}\frac{x^{z-1}}{e^{x}-1}dx,\
\operatorname{Re}(z)>1$

and

$\displaystyle\zeta(z)=2^{z}\pi^{z-1}\sin(\frac{z\pi}{2})\Gamma(1-z)\zeta(1-z),\
\operatorname{Re}(z)>1.$

One may extend analytic region of Riemann zeta function from
$\operatorname{Re}(z)>1$ to $\operatorname{Re}(z)>0$.

$\displaystyle\left(1-2^{1-z}\right)\zeta(z)=\sum_{n=1}^{\infty}\frac{(-1)^{n-1}}{n^{z}},\
\operatorname{Re}(z)>0\ \mbox{and}\ z\neq 1.$ (2)

For more classical representations of the Riemann zeta function, see [1, 2].
Here, we are aiming to present some new representations that may attract to
interested readers.

##  2 Main results

For sufficiently large $n$, one rewrites Riemann zeta function as

$\displaystyle\begin{aligned}
\zeta(z)&=1+\frac{1}{2^{z}}+\frac{1}{2^{2z}}+\frac{1}{2^{3z}}+\frac{1}{2^{4z}}+\cdots\\\
&+\frac{1}{3^{z}}+\frac{1}{3^{2z}}+\frac{1}{3^{3z}}+\frac{1}{3^{4z}}+\cdots\\\
&+\frac{1}{5^{z}}+\frac{1}{5^{2z}}+\frac{1}{5^{3z}}+\frac{1}{5^{4z}}+\cdots\\\
&+\frac{1}{6^{z}}+\frac{1}{6^{2z}}+\frac{1}{6^{3z}}+\frac{1}{6^{4z}}+\cdots\\\
&+\cdots+\frac{1}{n^{z}}+\frac{1}{n^{2z}}+\frac{1}{n^{3z}}+\frac{1}{n^{4z}}+R_{n}(z)\end{aligned}$
(3)

where $n\neq 2^{k},3^{k},5^{k},6^{k},7^{k},10^{k},11^{k},12^{k},\cdots$ with
$k=2,3,4,\cdots$ and $R_{n}(z)$ is the remainder. It is important to note that
these excluded values are already included in (3). So by applying geometric
series to (3) for sufficiently large $n$, one obtains $n$th approximation to
the Riemann zeta function as

$\displaystyle\begin{aligned}
\zeta(z)&=\frac{1}{1-\frac{1}{2^{z}}}+\left(\frac{1}{1-\frac{1}{3^{z}}}-1\right)+\left(\frac{1}{1-\frac{1}{5^{z}}}-1\right)+\left(\frac{1}{1-\frac{1}{6^{z}}}-1\right)\\\
&+\left(\frac{1}{1-\frac{1}{7^{z}}}-1\right)+\left(\frac{1}{1-\frac{1}{10^{z}}}-1\right)+\left(\frac{1}{1-\frac{1}{11^{z}}}-1\right)\\\
&+\left(\frac{1}{1-\frac{1}{12^{z}}}-1\right)+\cdots+\left(\frac{1}{1-\frac{1}{n^{z}}}-1\right).\\\
\end{aligned}$ (4)

Here, $R_{n}(z)$ goes to zero for sufficiently large $n$. Indeed, if
$\operatorname{Re}(z)=\sigma>1$ then

$\displaystyle\lvert
R_{n}(z)\rvert\leq\frac{1}{(n+1)^{\sigma}}+\frac{1}{(n+2)^{\sigma}}+\frac{1}{(n+3)^{\sigma}}+\cdots.$

So, $\lvert R_{n}(z)\rvert\to 0$ for $n\to\infty$. After arranging (4), one
obtains $n$th approximation as

$\displaystyle\begin{aligned}
\zeta(z)&=1+\frac{1}{2^{z}-1}+\frac{1}{3^{z}-1}+\frac{1}{5^{z}-1}+\frac{1}{6^{z}-1}+\frac{1}{7^{z}-1}\\\
&+\frac{1}{10^{z}-1}+\frac{1}{11^{z}-1}+\frac{1}{12^{z}-1}+\cdots+\frac{1}{n^{z}-1}.\\\
\end{aligned}$ (5)

One may rewrite (5) as series

$\displaystyle\begin{aligned}
\zeta(z)=1+\sum_{r=2}^{n}\frac{1}{r^{z}-1}\end{aligned}$ (6)

where $r\neq 2^{k},3^{k},5^{k},6^{k},7^{k},10^{k},11^{k},12^{k},\cdots$ with
$k=2,3,4,\cdots$.

$\displaystyle\frac{1}{e^{z}-1}+\frac{1}{2}=\frac{1}{2}\coth\left(\frac{z}{2}\right).$
(7)

It is well known that the above identity has a very important role in Riemann
zeta function. So one also rewrites (6) by using (7) as

$\displaystyle\begin{aligned}
\zeta(z)&=1+\frac{1}{2}\overbrace{\sum_{r=2}^{n}\coth\left(\frac{z\log
r}{2}\right)}^{\text{$l=$ number of terms}}-\frac{l}{2}\\\
&=\frac{2-l}{2}+\frac{1}{2}\sum_{r=2}^{n}\coth\left(\frac{z\log
r}{2}\right)\\\ \end{aligned}$ (8)

where $r\neq 2^{k},3^{k},5^{k},6^{k},7^{k},10^{k},11^{k},12^{k},\cdots$ with
$k=2,3,4,\cdots$. So one formally has the following theorems.

######  Theorem 1.

Let (1) hold. Then the following are equivalent.

  * i.

(5) is true.

  * ii.

(8) is true.

######  Theorem 2.

Let $r\neq 2^{k},3^{k},5^{k},6^{k},7^{k},10^{k},11^{k},12^{k},\cdots$ be where
$k=2,3,4,\cdots$. Then (5) has the following series representation.

$\displaystyle\begin{aligned}
\zeta(z)&=\sum_{n=-1}^{\infty}\frac{z^{n}B_{1+n}\sum_{r=2}^{n}\log^{n}r}{(1+n)!}\\\
&=\sum_{n=-1}^{\infty}\frac{z^{n}B_{1+n}\left(\log^{n}2+\log^{n}3+\cdots+\log^{n}n\right)}{(1+n)!}.\end{aligned}$
(9)

Here, $B_{1+n}$ is $(1+n)$ th Bernoulli numbers [11] and $r$ is defined as
above. One may study zeros of partial sums (9) similar to [14, 15, 16].

From (5) and (8), one gets the following approximation where $m$ is a positive
integer.

######  Corollary 1.

Let $r\neq 2^{k},3^{k},5^{k},6^{k},7^{k},10^{k},11^{k},12^{k},\cdots$ be where
$k=2,3,4,\cdots$. Then

$\displaystyle\begin{aligned}
\zeta(m)&=1+\overbrace{\sum_{r=2}^{n}\frac{1}{r^{m}-1}}^{\text{$l=$ number of
terms}}\\\ &=\frac{2-l}{2}+\frac{1}{2}\sum_{r=2}^{n}\coth\left(\frac{m\log
r)}{2}\right).\end{aligned}$ (10)

######  Corollary 2.

Let $r\neq 2^{k},3^{k},5^{k},6^{k},7^{k},10^{k},11^{k},12^{k},\cdots$ be where
$k=2,3,4,\cdots$. Then

$\displaystyle\begin{aligned}
\zeta(2m)&=1+\overbrace{\sum_{r=2}^{n}\frac{1}{r^{2m}-1}}^{\text{$l=$ number
of terms}}\\\ &=\frac{2-l}{2}+\frac{1}{2}\sum_{r=2}^{n}\coth\left(m\log
r\right).\end{aligned}$ (11)

The values $\zeta(2m)$ are well known due to Euler [12] where $m$ is an
integral.

######  Corollary 3.

Let $r\neq 2^{k},3^{k},5^{k},6^{k},7^{k},10^{k},11^{k},12^{k},\cdots$ be where
$k=2,3,4,\cdots$. Then

$\displaystyle\begin{aligned}
\zeta(2m+1)&=1+\overbrace{\sum_{r=2}^{n}\frac{1}{r^{2m+1}-1}}^{\text{$l=$
number of terms}}\\\
&=\frac{2-l}{2}+\frac{1}{2}\sum_{r=2}^{n}\coth\left((m+\frac{1}{2})\log
r\right).\end{aligned}$ (12)

The values $\zeta(2m+1)$ are not known yet except $\zeta(3)$ [13] however some
representations of $\zeta(2m+1)$ are given in terms of special functions.

One may want to know the zeros of the $n$th approximations obtained above. We
may at least reach this goal by using the program in [18]. By using ”solve”
command, one obtains the followings.

$\displaystyle 1+\frac{1}{2^{x}-1}=0$

There are not any solutions. It is also easy to see that without using any
program since it is monotonically decreasing.

$\displaystyle 1+\frac{1}{2^{x}-1}+\frac{1}{3^{x}-1}=0$

There are two complex solutions $x\approx-5.30989\times 10^{-17}\pm
3.50671i\dots$

$\displaystyle 1+\frac{1}{2^{x}-1}+\frac{1}{3^{x}-1}+\frac{1}{5^{x}-1}=0$

There are two complex solutions $x\approx 0.445959\pm 2.81436i\dots$

$\displaystyle
1+\frac{1}{2^{x}-1}+\frac{1}{3^{x}-1}+\frac{1}{5^{x}-1}+\frac{1}{6^{x}-1}=0$

There are two complex solutions $x\approx 0.631214\pm 2.54663i\dots$.

By applying the same argument as in (3). One rewrites (2) for sufficiently
large $n$ as

$\displaystyle\begin{aligned}
\left(1-2^{1-z}\right)\zeta(z)&=\Big{(}1+\frac{1}{3^{z}}+\frac{1}{3^{2z}}+\frac{1}{3^{3z}}+\frac{1}{3^{4z}}+\cdots\\\
&+\frac{1}{5^{z}}+\frac{1}{5^{2z}}+\frac{1}{5^{3z}}+\frac{1}{5^{4z}}+\cdots\\\
&+\frac{1}{7^{z}}+\frac{1}{7^{2z}}+\frac{1}{7^{3z}}+\frac{1}{7^{4z}}+\cdots\Big{)}\\\
&+\cdots-\Big{(}\frac{1}{2^{z}}+\frac{1}{2^{2z}}+\frac{1}{2^{3z}}+\frac{1}{2^{4z}}+\cdots\\\
&+\frac{1}{6^{z}}+\frac{1}{6^{2z}}+\frac{1}{6^{3z}}+\frac{1}{6^{4z}}+\cdots\\\
&+\frac{1}{10^{z}}+\frac{1}{10^{2z}}+\frac{1}{10^{3z}}+\frac{1}{10^{4z}}+\cdots\Big{)}\\\
\end{aligned}$ (13)

So by applying geometric series to (13) as before one obtains

$\displaystyle\begin{aligned}
\left(1-2^{1-z}\right)\zeta(z)&=1+\frac{1}{3^{z}-1}-\frac{1}{2^{z}-1}+\frac{1}{5^{z}-1}-\frac{1}{6^{z}-1}\\\
&+\frac{1}{7^{z}-1}-\frac{1}{10^{z}-1}+\frac{1}{11^{z}-1}-\frac{1}{12^{z}-1}\\\
&+\frac{1}{13^{z}-1}-\frac{1}{14^{z}-1}+\cdots+R_{n}(z).\end{aligned}$ (14)

As before, $\lvert R_{n}(z)\rvert\to 0$ as $n\to\infty$. One rewrites $n$th
appriximation to (14) as

$\displaystyle\begin{aligned}
\zeta(z)=\frac{1}{1-2^{1-z}}\left(1+\sum_{r=2}^{n}(-1)^{r-1}\frac{1}{r^{z}-1}\right)\end{aligned}$
(15)

where $r\neq 2^{k},3^{k},5^{k},6^{k},7^{k},10^{k},11^{k},12^{k},\cdots$ and
$k=2,3,4,\cdots$. Once again by using (7), one obtains the following
representation to Riemann zeta function for sufficiently large $n$ as

$\displaystyle\zeta(z)=\begin{cases}\frac{1}{1-2^{1-z}}\left(1+\frac{1}{2}\overbrace{\sum_{r=2}^{n}(-1)^{r-1}\coth\left(\frac{z\log
r}{2}\right)}^{\text{$l=$ number of terms is even}}\right),&\text{if $n$ is
even}\\\
\frac{1}{1-2^{1-z}}\left(\frac{1}{2}+\frac{1}{2}\overbrace{\sum_{r=2}^{n}(-1)^{r-1}\coth\left(\frac{z\log
r}{2}\right)}^{\text{$l=$ number of terms is odd}}\right),&\text{if $n$ is
odd}.\\\ \end{cases}$ (16)

So once again one formally has the following theorems.

######  Theorem 3.

Let (2) hold. Then the following are equivalent.

  * i.

(15) is true.

  * ii.

(16) is true.

######  Remark 1.

Similar corollaries may be derived related $\zeta(m)$, $\zeta(2m)$ and
$\zeta(2m+1)$ as stated in Corollary 1, 2 and 3.

These $n$th approximations gives the following numerical results.

$\displaystyle 1-\frac{1}{2^{x}-1}=0$

There are infinitely many complex solutions $x=1+\frac{2i\pi n}{\log 2}$ where
$n$ is an integer numbers. It is easy to obtain these roots by solving it
directly. $2^{x-1}=1$. From here, by using $\log z$ where $z$ is a complex
number. One gets, $x=1+\frac{2i\pi n}{\log 2}$. If

$\displaystyle 1-\frac{1}{2^{x}-1}+\frac{1}{3^{x}-1}=0,$

then here is only one real solution $x\approx 0.523205\dots$ If

$\displaystyle\frac{1}{2}-\frac{1}{2^{x}-1}+\frac{1}{3^{x}-1}+\frac{1}{5^{x}-1}=0,$

then there are two complex solutions $x\approx-1.06986\times 10^{-15}\pm
0.719409i\dots$ If

$\displaystyle
1-\frac{1}{2^{x}-1}+\frac{1}{3^{x}-1}+\frac{1}{5^{x}-1}-\frac{1}{6^{x}-1}=0,$

then there is only one real solution $x\approx 0.465171\dots$

##  3 Conclusion

To the best of author’s knowledge this approach applied to Riemann zeta
function is not reported in the existing literature yet. The author thinks
that Theorem 1 and 3 are good approximations to Riemann zeta function for
sufficiently large $n$. Part of this article is presented as a talk in [17].
It would be very nice to see more zeros of the $n$th approximations however
the program in [18] does not run any more. After adding more terms, computer
runs into word saying ”standard computation time exceeded”.

## References

  * [1] E. C. Titchmarsh, The Zeta Function of Riemann, Cambridge University Press, Cambridge, 1930.
  * [2] E. C. Titchmarsh, The Theory of the Riemann Zeta-Function, Oxford University Press, Oxford, 1986.
  * [3] H. M. Edwards, Riemann’s Zeta Function, Academic Press, New York, 1974.
  * [4] S. J. Patterson, An Introduction to the Theory of the Riemann Zeta-Function, Cambridge University Press, Cambridge, 1988.
  * [5] Exploring the Riemann Zeta Function, 190 years from Riemann’s Birth, ed. by H. Montgomery, A. Nikeghbali, M. Th. Rassias, Springer, Berlin, 2017.
  * [6] K. Chandrasekharan, Lectures on the Riemann Zeta-Function, Tata Institute of Fundamental Research, Bombay, India, 1958.
  * [7] B. Riemann, Über die Anzahl der Primzahlen unter einer gegebenen Grösse, Monatsber. Akad., Berlin, 671–680. 1859.
  * [8] H. Iwaniec, Lectures on the Riemann Zeta Function, vol. 62 of University Lecture Series, American Mathematical Society, Providence, RI, 2014.
  * [9] Don Bernard Zagier, Zeta Functions in Number Theory, Annual Meeting of the American Mathematics Society, Phoenix, Arizona, January 1989.
  * [10] E. Elizalde, Ten Physical Applications of Spectral Zeta Functions, Springer-Verlag, Berlin, Germany, 1995.
  * [11] T. Arakawa, T. Ibukiyama and M. Kaneko, Bernoulli Numbers and Zeta Functions, Springer, Japan, 2014.
  * [12] L. Euler, De numero memorabili in summatione progressionis harmonicae naturalis occurrente, Opera Omnia, series prima XV (1785), 567–603
  * [13] R. Apéry, Interpolation de fractions continues et irrationalite de certaines constantes, Bull. Section des Sci., Tome III, Bibliothque Nationale, Paris, 1981, 37-63.
  * [14] C. Y. Yildirim, A sum over zeros of partial sums of $e^{x}$, J. Ramanujan Math. Soc. 6 (1991), 51–66.
  * [15] C. Y. Yildirim, On the zeros of the sections of the exponential function, Turk. J. Math. 3 (1992), 177-182.
  * [16] C. Y. Yildirim, On the tails of the exponential series, Canad. Math. Bull. 37 (1994), 278-286.
  * [17] T. Tanriverdi, Notes on the Riemann Zeta Function, 7th International Conference on Applied Analysis and Mathematical Modeling (ICAAMM18), İstanbul Gelişim Üniversitesi, İstanbul, 20-24 June 2018.
  * [18] https://www.wolframalpha.com


# Crow instability of vortex lines in dipolar superfluids

Srivatsa B. Prasad<EMAIL_ADDRESS>Joint Quantum Centre
Durham-Newcastle, School of Mathematics, Statistics and Physics, Newcastle
University, Newcastle upon Tyne, NE1 7RU, United Kingdom  Nick G. Parker
<EMAIL_ADDRESS>Joint Quantum Centre Durham-Newcastle, School of
Mathematics, Statistics and Physics, Newcastle University, Newcastle upon
Tyne, NE1 7RU, United Kingdom  Andrew W. Baggaley
<EMAIL_ADDRESS>Joint Quantum Centre Durham-Newcastle, School
of Mathematics, Statistics and Physics, Newcastle University, Newcastle upon
Tyne, NE1 7RU, United Kingdom

###### Abstract

In classical inviscid fluids, parallel vortices perturbed by Kelvin waves may
exhibit the Crow instability, where the mutual interaction of the Kelvin modes
renders them dynamically unstable. This results in the mutual approach and
reconnection of the vortices, leading to a cascaded decay into smaller and
smaller vortex loops. We study the Crow instability of quantum vortex lines in
a superfluid enjoying the anisotropic, long-ranged dipole-dipole interaction
through mean-field simulations of the dynamics of the superfluid order
parameter. We observe that both the strength and direction of the dipole-
dipole interaction play a crucial role in determining the Kelvin modes that
are dynamically favoured. This is shown to be linked to effective enhancements
or suppressions of the vortices’ curvature and can be explained by the
effective dipole-dipole interaction between the vortex lines themselves. This
paves the way to a deeper understanding of vortex reconnection dynamics,
vortex loop cascading and turbulence in dipolar superfluids.

##  I Introduction

Classical inviscid fluids play host to a variety of instabilities arising from
the interaction of waves and vortices, and understanding their properties is a
central tenet of fluid dynamics. One such example is the celebrated Crow
instability, where a pair of vortices with mutually antiparallel circulations
is unstable against transverse symmetric perturbations that induce helical
Kelvin waves upon the vortex tubes. These Kelvin waves grow in amplitude until
the vortices reconnect, thereby forming a series of vortex loops. The
subsequent cascade of reconnections to ever-smaller vortex loops eventuates in
dissipation of the loops, as seen in the mitigation of wingtip vortices of
aircraft by the interaction with the corresponding contrails [1]. The
existence and properties of analogues of such hydrodynamic instabilities has
proved to be a fruitful source of inspiration for studying quantum fluids,
such as Bose-Einstein condensates, that exhibit superfluidity. Unlike
classical fluids described theoretically by the Navier-Stokes equations,
superfluids can flow without viscous dissipation and are characterised by a
locally coherent order parameter [2]. The phase coherence of this order
parameter ensures that the vorticity of a superfluid can only be nonzero along
discrete, infinitesimally thin lines, each a topological defect, such that
each vortex line boasts a quantised circulation and a total depletion of the
superfluid density at its core [3]. As such, dynamical processes involving
quantum vortices, such as reconnections and annihilations, [4, 5] cannot
change their total net circulation as it is a topologically conserved quantity
[3]. Whereas the ground state of an ensemble of quantum vortices – typically,
a triangular vortex lattice – is well-understood [6, 7, 8, 9], the interwoven
phenomena of instabilities, turbulence and disorder of incompressible
(vortices) and compressible (phonon) excitations in superfluids have been of
great interest for some time [10, 11, 12, 13, 14]. While certain superfluid
instabilities such as the snake instability [15] have no classical counterpart
it has been shown that superfluids, or mixtures thereof, can become turbulent
through quantum analogues of classical hydrodynamic instabilities such as the
Kelvin-Helmholtz [16, 17, 18], Rayleigh-Taylor [19, 20] and Richtmyer-Meshkov
[21] instabilities. The Crow instability, too, manifests itself in the
dynamics of pairs of quantum vortices of opposite circulations when subjected
to either an environmental perturbation or if there are pre-existing Kelvin
waves on the vortex lines [15, 22, 23, 24, 25]. An example of the latter
scenario is depicted in Fig. 1, showing how the magnitude of the transverse
excitations of the vortex lines grows in time till the vortices’ cores overlap
at a point in space to the extent that they are able to reconnect into a pair
of loops 111The method by which this figure is obtained is presented later in
this article..

Figure 1: The superfluid Crow instability of an antiparallel quantum vortex
pair initially separated along the $y$-axis. With $t_{\mathrm{r}}$ the time
till the first reconnection, the vortices are depicted at
$t=t_{\mathrm{r}}/3$, $t=2t_{\mathrm{r}}/3$ and $t=t_{\mathrm{c}}\equiv
t_{\mathrm{r}}-0.5$.

The vast majority of investigations of hydrodynamic instabilities in
superfluids have focussed on systems where the interactions between its
constituents can be approximated as being effectively isotropic and short-
ranged. Dipolar Bose-Einstein condensates (dBECs), comprised of certain
lanthanide atoms with large, permanent magnetic dipole moments, represent a
counterpoint to this paradigm. When its atomic dipole moments are uniformly
polarized by an applied magnetic field, a dBEC exhibits a wealth of exotic
phenomena that arise from a delicate interplay between the long-ranged and
anisotropic dipole-dipole interaction (DDI) and the residual short-ranged,
effectively isotropic van der Waals repulsion [26, 27, 28]. The foremost
consequence of this competition is the tendency of a dBEC to exhibit
anisotropy, whether it be its density profile under external confinement [29,
30], the speed of sound and superfluid critical velocity [31, 32], or the core
of an embedded quantum vortex [33, 34, 35, 36]. Another consequence is a
tendency towards stratification or short-ranged order, often associated with
the presence of intermediate-wavelength roton excitations, resulting in the
manifestation of novel features such as non-triangular vortex lattices with
density striping [37, 38, 27, 39, 35], density ripples about the cores of
quasi-two-dimensional point vortices [34], stratification during turbulence
[40], pattern formation [41, 42], and the hitherto elusive supersolid phase
[43, 44, 45]. Theoretical studies of quantum vortex dynamics in dBECs, which
are now of greater relevance than ever after the experimental realization of
dipolar quantum vortices [46, 47], have also demonstrated evidence for
considerable deviations away from the nondipolar limit of properties such as
the velocities and trajectories of vortex pairs [34, 48, 49, 36, 50] and the
Kelvin wave spectra of single vortices [51, 52]. Together, these results would
suggest that the onset of a hydrodynamic instability such as the Crow
instability is sensitive to the magnitude and direction of the dipole moment
polarization.

Thus, in this article we present the results of a systematic analysis of the
Crow instability manifesting itself in a dBEC with a uniform background
density and elucidate the ways in which the various properties of the
instability depend on the parameters of the DDI. In Section II we provide an
overview of the dipolar Gross-Pitaevskii equation (dGPE) as employed to model
perturbed quantum vortices in a uniform, three-dimensional dBEC. We also
discuss the numerical methods used for propagating the dGPE and identifying
vortex structures in the solutions. In Sec. III we introduce a scenario where
each of a pair of antiparallel vortices is subjected to a symmetric, random
perturbation and provide a qualitative discussion of the results of one such
simulation, thereby illustrating the influence that the DDI has upon the
vortex profiles that emerge when the Crow instability is triggered. Section IV
presents a spectral study of these vortex profiles, where tracking the
populations of each Kelvin wave mode allows us to identify the modes that are
excited most strongly by the Crow instability, and their corresponding rates
of amplitude growth, as well as the line-averaged curvature of the vortices.
These findings are summarised in Sec. V along with some implications and
possible generalisations of this work.

##  II Formalism

In this article, we investigate the dynamics of perturbed pairs of quantum
vortices in a dBEC through propagating the dipolar Gross-Pitaevskii equation
(dGPE) governing the superfluid order parameter, $\psi(\mathbf{r},t)$. For a
system composed of a single atomic species of mass $m$ and magnetic dipole
moment $\mu_{\mathrm{d}}$, polarised uniformly by a magnetic field parallel to
$d$, the dGPE is given by [26, 27, 28],

$i\hbar\frac{\partial\psi}{\partial
t}=\left\\{-\frac{\hbar^{2}}{2m}\nabla^{2}+\frac{4\pi\hbar^{2}a_{\mathrm{s}}}{m}\left[\delta(\mathbf{r})+3\varepsilon_{\mathrm{dd}}V_{\mathrm{dd}}\right]\circledast
n-\mu\right\\}\psi.$ (1)

The dGPE encapsulates the effects of both the short-ranged two-body
interaction, mediated by $a_{\mathrm{s}}$, the scattering length of the atom-
atom scattering potential, and the dipole-dipole interaction (DDI), which is
defined in real and reciprocal space as [26]

$\displaystyle V_{\mathrm{dd}}(\mathbf{r})$
$\displaystyle=\frac{1}{4\pi}\left[\frac{1-3(\hat{\mathbf{B}}\cdot\hat{\mathbf{r}})^{2}}{r^{3}}\right],$
(2) $\displaystyle\widetilde{V}_{\mathrm{dd}}(\mathbf{q})$
$\displaystyle=(\hat{\mathbf{B}}\cdot\hat{\mathbf{q}})^{2}-\frac{1}{3},$ (3)

respectively. We represent the ratio between the interaction strengths of the
DDI and the short-ranged interaction through the parameter
$\varepsilon_{\mathrm{dd}}=m\mu_{0}\mu_{\mathrm{d}}^{2}/(12\pi\hbar^{2}a_{\mathrm{s}})$,
where $\mu_{0}$ is the permeability of free space. Additionally, in Eq. (1)
the superfluid’s density and chemical potential are represented as
$n=|\psi|^{2}$ and $\mu$, respectively.

As encapsulated by Eq. (2), the DDI between any two atoms in the dBEC is
repulsive (attractive) when the angle between them,
$\arccos(\hat{\mathbf{B}}\cdot\hat{\mathbf{r}})$, is less (more) than the
critical angle $\theta_{\mathrm{c}}=\arccos(1/\sqrt{3})\approx
54.7\,\mathrm{deg}$. The resulting tendency of the superfluid atoms to realign
themselves to minimise their dipolar energy is responsible for
magnetostrictive effects where the condensate density exhibits anisotropy
where a nondipolar condensate would not. When $\varepsilon_{\mathrm{dd}}>1$,
the solutions of Eq. (1) are unstable to collapse as the anisotropic mutual
dipolar attraction of the atoms overwhelms the short-ranged repulsion,
necessitating the inclusion of a beyond-mean-field energy correction [53, 54,
55] in such regimes to account for the existence and stability of exotic
states such as quantum droplets [56, 57] and supersolids [43, 44, 45]. For the
sake of simplicity, we focus on the regime $\varepsilon_{\mathrm{dd}}<1$,
where no such correction is necessary [28]. This renders it convenient to work
in natural units where energies are scaled by the ground state value of the
chemical potential, $\mu_{\mathrm{g}}=4\pi\hbar^{2}a_{\mathrm{s}}n_{0}/m$, for
a uniform background density $n_{0}$. Simultaneously, space and time are
scaled in units of the healing length $\xi=\hbar/\sqrt{m\mu_{\mathrm{g}}}$,
and $\hbar/\mu_{\mathrm{g}}$, respectively, while the order parameter is
scaled by $\sqrt{n_{0}}$ [58].

Our results are based on numerical solutions of the dGPE obtained by
propagation via the split-step pseudospectral method [59]. Throughout this
article, we work in a domain with dimensions
$\\{L_{x},\,L_{y},\,L_{z}\\}=\\{100\xi,\,100\xi,\,200\xi\\}$ along the $x$,
$y$\- and $z$-axes, respectively, corresponding to a spatial grid of
$256\,\times\,256\,\times\,512$ points. The use of discrete Fourier transforms
in the pseudospectral scheme to calculate spatial derivatives results in
periodic boundary conditions being imposed upon the solutions of Eq. (1). The
dGPE admits quantum vortex solutions, characterised by a quantized circulation
[3], viz.

$\Gamma=\oint\,\mathrm{d}\mathbf{s}\cdot\mathbf{v}=2\pi q\,:\,q\in\mathbb{Z},$
(4)

of the superfluid velocity,
$\mathbf{v}=\nabla\left[\mathrm{Im}(\log\psi)\right]$, around the vortex core.
As $\psi$ is singly-valued, this implies that the superfluid density vanishes
at the vortex core. To efficiently identify regions of the spatial domain
where the superfluid is characterized by both a quantized circulation and a
vanishing superfluid density and a quantized circulation, we compute the
superfluid’s pseudovorticity [60],

$\boldsymbol{\omega}_{\mathrm{ps}}=\frac{1}{2}\nabla\times\left(n\mathbf{v}\right)\equiv\nabla\mathrm{Re}[\psi]\times\nabla\mathrm{Im}[\psi].$
(5)

This quantity is only nonzero in the vicinity of a quantum vortex core and has
thus been used extensively in recent studies of superfluid vortex dynamics to
locate and track vortices [25, 61, 62, 63]. Subsequently, we employ the
Newton-Raphson method to obtain subgrid interpolations of the space curve
defining each vortex line in the domain.

##  III Evolution of the Vortex Lines

In a realistic experimental scenario, the Crow instability of two antiparallel
vortices would arise from an initial perturbation by a source of compressible
energy, where vortex-sound interactions seed Kelvin waves oscillations along
the vortices [1]. In a trapped Bose-Einstein condensate, one such scenario
would occur when a pair of vortices is located in a region where the trapping
potential landscape is not flat, thereby resulting in background density
gradients that perturb the vortices [23, 24]. In this article we have no such
external perturbation and, instead, the perturbation is implicit in the
initial conditions of the vortices [15, 22, 25]. Thus, we initialize our
simulations with the superfluid phase of an unperturbed pair of straight,
parallel vortex lines with mutually opposed circulations and apply a random
perturbation to the loci of the vortex lines along their respective lengths.
Let us assume without loss of generality that the two respective vorticities
of the unperturbed vortices are (anti-)parallel to the $z$-axis. Noting that
the phase of $\psi$, $S$, is effectively a superfluid velocity potential, i.e.
$\mathbf{v}=\nabla S$, let us recall the classical expression for the velocity
potential generated by a pair of straight, antiparallel vortices in an
incompressible fluid as defined in the periodic domain
$x\in[0,L_{x}),\,y\in[0,L_{y})$ [64, 65, 66]:

$\displaystyle S(x,y)=\sum_{p=-\infty}^{\infty}\biggl{\\{}$
$\displaystyle\mp\operatorname{atan}\left[\tanh\left(\frac{\pi
Y_{\pm}}{L_{y}}+p\pi\right)\tan\left(\frac{\pi
X_{\pm}}{L_{y}}-\frac{\pi}{2}\right)\right]$
$\displaystyle+\pi\left[\Theta(X_{+})-\Theta(X_{-})\right]\biggr{\\}}-\frac{2\pi(x_{+}-x_{-})y}{L_{x}L_{y}},$
(6)

Here, $(X_{\pm},Y_{\pm})=(x-x_{\pm},y-y_{\pm})$ where the vortex with
circulation $\pm 2\pi$ is located at $(x_{\pm},y_{\pm})$ 222In practice,
replacing the infinite sum over $p$ with a partial sum from $p=-P$ to $p=P$
results in a rapid convergence of $S(x,y)$ for small positive $P$; in this
investigation we have fixed $P=11$ in line with our previous work.. Let us
define the following random perturbation profile,

$\delta w(z)=\sum_{q=-20}^{20}\exp\left[i\pi(2\eta_{q}-1)+\frac{2\pi
iqz}{L_{z}}\right]-\exp\left[i\pi(2\eta_{0}-1)\right],$ (7)

representing the lowest $40$ excited modes along the $z$-axis equally
populated with a random phase $\eta_{q}$ drawn from the uniform distribution
in the interval $[0,1)$. The initial phase of $\psi$ in our simulations of the
Crow instability is then given by Eq. (6) with
$(x_{\pm},y_{\pm})\mapsto[x_{\pm}-\mathrm{Re}(\delta
w),y_{\pm}-\mathrm{Im}(\delta w)]$. Starting from a spatially uniform profile
as an initial ansatz for the superfluid density, the dGPE is evolved in
imaginary time, viz. under the substitution $t\mapsto it$, with our phase
ansatz held fixed until convergence of $\psi$ is achieved.

In the dimensionless dGPE theory there exist two independent parametric
degrees of freedom: $\mathbf{B}$ and $\varepsilon_{\mathrm{dd}}$. In our work
we consider $\varepsilon_{\mathrm{dd}}\in\\{0,0.1,0.2,\mathellipsis
0.8,0.9\\}$. This allows us to investigate the emergence of the Crow
instability across a range of regimes from the nondipolar limit to just below
the upper limit of the range of validity of the mean-field theory. For each
value of $\varepsilon_{\mathrm{dd}}$, the initial unperturbed separation of
the vortices in the $x$-$y$ plane is taken to be $d=6.25\xi$ along the
$y$-axis and $\mathbf{B}$ is allowed to be (anti)parallel to each vortex line
($\mathbf{B}\parallel\hat{z}$), parallel to the mean vortex separation
($\mathbf{B}\parallel\hat{y}$) or parallel to the net velocity of the vortex
pair ($\mathbf{B}\parallel\hat{x}$). For each choice of $\mathbf{B}$ and
$\varepsilon_{\mathrm{dd}}$, we have conducted simulations of the Crow
instability over an ensemble consisting of $4$ randomly selected sets of the
initial Kelvin wave phase profile, $\\{\eta_{q}\\}$.

Figure 2: The Crow instability depicted for $\varepsilon_{\mathrm{dd}}=0.9$
and $\mathbf{B}$ parallel to $\hat{x}$ (a), $\hat{y}$ (b) and $\hat{z}$ (c).

First, let us inspect the profiles of the vortex filaments during the
evolution of the dGPE. In Fig. 1 we presented snapshots of two vortex lines
exhibiting the Crow instability in the nondipolar limit. Below, in Fig. 2, we
depict the corresponding snapshots of the vortex pair with the same initial
conditions but with $\varepsilon_{\mathrm{dd}}=0.9$ and $\mathbf{B}$ being
parallel to $\hat{x}$ (first row), $\hat{y}$ (second row), or $\hat{z}$ (third
row). In each of these subplots, the vortices are plotted at the times
$t=t_{\mathrm{r}}/3$, $t=2t_{\mathrm{r}}/3$ and $0.5$ units of dimensionless
time before the first reconnection occurs at $t=t_{\mathrm{r}}$; for the sake
of brevity we denote this value of $t$ as $t_{\mathrm{c}}=t_{\mathrm{r}}-1/2$.
Note that $t_{\mathrm{r}}$, and thus $t_{\mathrm{c}}$, is a nontrivial
function of $\mathbf{B}$ and $\varepsilon_{\mathrm{dd}}$ as well as initial
conditions, and that the translation speed of the mean orientation of the
vortex pair is also a function of $\mathbf{B}$ and $\varepsilon_{\mathrm{dd}}$
[36], such that in each row $t_{\mathrm{r}}$ attains a different value and the
vortices are displaced by different distances along the $y$-axes from their
initial positions.

Figure 2 highlights that the final vortex profiles shortly before the
reconnection induced by the Crow instability are strongly dependent on the
direction of the dipolar polarisation relative to the orientation of the
vortices. Compared to the corresponding vortex lines in the nondipolar limit
in Fig. 1, the vortex lines when $\mathbf{B}\parallel\hat{z}$ are noticeably
straighter for each of the three times as depicted in Fig. 1 (c); whereas the
nondipolar limit is characterised by a highly agitated pair of vortex lines
when $t=t_{\mathrm{c}}$ with multiple points along each vortex line where the
vortices are curved towards each other and appear to heading for reconnection,
only one such point is evident in the $z$-polarized case. The opposite case
occurs for a dipole polarization along the $y$-axis, which is parallel to the
mean separation of the vortices and is depicted in Fig. 2 (b). In this case,
while the first reconnection occurs at approximately the same location along
the $z$-axis as in the nondipolar limit, the spatial separation of the
vortices at the other two separation minima is appreciably smaller, suggesting
that the vortices would undergo their second and third reconnections sooner
than when $\varepsilon_{\mathrm{dd}}=0$. The third case, where
$\mathbf{B}\parallel\hat{z}$, is distinguishable from both of the other
polarizations as the points where the vortex separation attains local minima
are characterised by larger displacements of each vortex from its line-
averaged mean position along the $x$-axis than for any of the other
polarizations; this is particularly evident when $t=t_{\mathrm{c}}$.

##  IV Properties of the Vortex Lines

In Sec. III we have shown that the profiles of antiparallel vortex lines
subjected to a small initial perturbation depend considerably on the strength
and orientation of the DDI experienced by the background dipolar superfluid.
This would suggest that the populations of the Kelvin waves that are excited
through the mechanism underlying the Crow instability are dependent in some
manner on the specifics of the DDI. In this section we investigate the
spectral properties of the vortex line profiles and characterise the Kelvin
wave populations during the simulations of the dGPE presented in the previous
section.

Let us proceed by representing the coordinates of the $n$th vortex line as
$w_{n}(z)=x_{n}(z)+iy_{n}(z)$. Shifting the average of these coordinates over
the $z$-axis to the origin of the $x$-$y$ plane yields the quantity
$\tilde{w}_{n}(z,t)=w_{n}(z,t)-\langle w_{n}\rangle(t)$, where
$\tilde{w}_{n}(z,t=0)=(-1)^{n}\delta w(z)$ is the initial Kelvin wave
perturbation given by Eq. (7). The corresponding mode amplitudes,
$W_{n}(k_{z},t)$, are the discrete Fourier transforms of $\tilde{w}_{n}(z,t)$
and the periodicity of the domain ensures a restriction of the mode
wavenumber, $k_{z}$, such that $k_{z}=2\pi q/L_{z},\,q\in\mathbb{Z}$. We also
note that, by construction, $W_{n}(k_{z}=0,t)=0$. Initially, we study the
temporal evolution of these mode amplitudes, focussing on those initially
occupied at $t=0$. Let us define the proportional amplitude of a mode,
$W^{\prime}(k_{z},t)$, as

$W^{\prime}(k_{z},t)=\frac{|W_{1}(k_{z},t)|+|W_{2}(-k_{z},t)|}{\sum_{k_{z}>0}\left[|W_{1}(k_{z},t)|+|W_{2}(-k_{z},t)|\right]}.$
(8)

Figure 3 depicts $W^{\prime}(k_{z},t)$, with $k_{z}$ restricted to the $20$
lowest-lying modes with positive wavenumber, for the nondipolar system (a) as
well as the maximally dipolar regime, $\varepsilon_{\mathrm{dd}}=0.9$, with
$\mathbf{B}$ parallel to $\hat{x}$ (b), $\hat{y}$ (c), or $\hat{z}$ (d). For
each distinct regime, we average the results over the $4$ sets of initial
conditions over the duration of time in which all $4$ simulations have not yet
undergone a vortex reconnection.

Figure 3: The relative occupation of the first 20 Kelvin wave modes with
positive $k_{z}$ during time evolution, averaged over the ensemble of initial
state Kelvin wave profiles. In (a), (top left), $\varepsilon_{\mathrm{dd}}=0$
and in the remaining subfigures, $\varepsilon_{\mathrm{dd}}=0.9$; $\mathbf{B}$
is parallel to $\hat{x}$ in (b), $\hat{y}$ (c) and $\hat{z}$ (d).

An inspection of Fig. 3 affords us the following insights into the evolution
of the Kelvin modes on the vortices. Regardless of the dipolar regime, we can
see that at early times the interaction of the modes results in fluctuations
of the relative mode populations, accompanied by the suppression of strongly
energetic high order modes with $p\gtrsim 10$. It is not until the vortices
enter their final stage of evolution – that is, with reference to the
snapshots of the vortex states in Figs. 1 and 2, when $t>2t_{\mathrm{r}}/3$ –
that unambiguous signatures of a dependence of the Kelvin mode population on
$\mathbf{B}$ and $\varepsilon_{\mathrm{dd}}$ become apparent. For large values
of $\varepsilon_{\mathrm{dd}}$ with the dipole moments polarised
(anti-)parallel to the vortex lines, as is the case in Fig. 3(d), the Kelvin
mode populations become monotonically decreasing as a function of $k_{z}$ as
$t\rightarrow t_{\mathrm{r}}$. A comparison with Fig. 2 (c) shows that these
spectral features are consistent with the qualitative appearance of the vortex
lines at late times since, for both vortices at $t=t_{\mathrm{c}}$, only one
pair of antinodes of the vortex displacement from its mean position is
apparent. By contrast, it is clear that polarising the dipole moments
orthogonal to the vorticity of either vortex stimulates the population of
higher Kelvin modes. The averaged spectrum for polarisations along the
translational axis of the vortex pair in Fig. 3(b) illustrates clearly that
the $q=3$ mode becomes overwhelmingly preferred as the Crow instability
manifests itself and the vortices approach their first reconnection. In line
with our observations for the $\mathbf{B}\parallel\hat{z}$ case, this aspect
of the Kelvin wave spectrum for the $\mathbf{B}\parallel\hat{x}$ case is
consistent with the plot of the vortices’ state at $t=t_{\mathrm{c}}$ in Fig.
2 (a) where $3$ distinct antinode pairs are distinguishable on each vortex.
Intriguingly, while Fig. 3 (c) shows that odd values of the Kelvin mode index
$q$ appear to be preferred as $t\rightarrow t_{\mathrm{c}}$ when the dipole
polarisation is parallel to the vortices’ mean separation, it is evident that
no single mode is overwhelmingly dominant in this regime and, similarly to the
nondipolar limit in Fig. 3 (a), the first $6$ modes contribute to the majority
of the mode population when $\mathbf{B}\parallel\hat{y}$. Indeed, while the
two antinodes of the vortices at $z\approx-20$ and $z\approx-50$ when
$t=t_{\mathrm{c}}$ appear to be closer to instigating reconnections in Fig. 2
(b) than the corresponding antinodes are in Fig. 1, the nondipolar vortices
are otherwise much more similar to the dipolar vortices in Fig. 2 (b) than
those in (a) and (c) when $t=t_{\mathrm{c}}$.

Figure 4: The wavenumber, $k_{z}$, of the Kelvin mode that is maximally
occupied by the vortex pair immediately prior to their mutual reconnection as
a function of $\varepsilon_{\mathrm{dd}}$ where the dipole polarisation is
parallel to $\hat{x}$ (a), $\hat{y}$ (b), and $\hat{z}$ (c). In each of these
subplots, distinct curves correspond to distinct Kelvin wave initial
conditions indexed by $\eta\in\\{1,\,2,\,3,\,4\\}$.

With these insights on the evolution of $W^{\prime}(k_{z},t)$ over all of the
low-lying modes for the maximally dipolar regime,
$\varepsilon_{\mathrm{dd}}=0.9$, we proceed to study specific features of the
mode spectrum for the full range of $\varepsilon_{\mathrm{dd}}$ from $0$ to
$0.9$. Initially, let us consider the maximally unstable Kelvin mode for a
given choice of $\\{\mathbf{B},\varepsilon_{\mathrm{dd}}\\}$, viz. the value
of $k_{z}$, denoted hereafter as $k_{\mathrm{c}}$, that maximises
$W^{\prime}(k_{z},t_{\mathrm{c}})$. Figure 4 presents plots of this wavenumber
as a function of $\varepsilon_{\mathrm{dd}}\in[0.0.9]$, with $\mathbf{B}$
parallel to $\hat{x}$ (a), $\hat{y}$ (b), and $\hat{z}$ (c), where each choice
of initial condition has been plotted as a distinct curve in each subplot. In
the nondipolar limit we find that, depending on the initial conditions, either
the 3rd or 5th Kelvin modes are most strongly excited through the Crow
instability. Taking the higher of the two modes and comparing its wavenumber,
$k_{\mathrm{c}}\xi=10\pi/200\approx 0.157$, to earlier studies of the
nondipolar Crow instability shows that this is consistent with predictions
that $dk_{c}\sim 1$ [22].

When the dipolar interaction strength is nonzero the behaviour of
$k_{\mathrm{c}}$ is dependent to the dipole polarisation relative to the
vortex configuration. When $\mathbf{B}\parallel\hat{y}$, $k_{\mathrm{c}}$ is
particularly insensitive to $\varepsilon_{\mathrm{dd}}$ and the specific mode
that is most strongly excited as a consequence of the Crow instability appears
to be determined only by our choice of initial conditions. This, we note, is
consistent with the ensemble-averaged relative mode populations in Fig. 3
being qualitatively similar for both the nondipolar limit (a) and for the
regime $\varepsilon_{\mathrm{dd}}=0.9,\,\mathbf{B}\parallel\hat{y}$ (c) as
$t\rightarrow t_{\mathrm{c}}$. However, such robustness against changes in the
relative dipolar interactions strength is not evident when the dipole moments
are polarised along either $\mathbf{x}$ or $\mathbf{z}$. In Fig. 2 (d), for
the regime $\varepsilon_{\mathrm{dd}}=0.9,\,\mathbf{B}\parallel\hat{z}$ it was
apparent that the $q=1$ Kelvin mode becomes overwhelmingly preferred as
$t\rightarrow t_{\mathrm{c}}$. This is supported by the behaviour of
$k_{\mathrm{c}}$ in Fig. 4 (c) for this regime where we see that only one of
the $4$ initial conditions results in the $q=2$ mode being most strongly
excited with the remainder preferentially occupying the $q=1$ mode. We also
find that, regardless of the initial conditions applied, $k_{\mathrm{c}}$ is a
decreasing function of $\varepsilon_{\mathrm{dd}}$ when
$\mathbf{B}\parallel\hat{z}$, such that the vortex loops formed as a
consequence of the Crow instability are unambiguously larger for more strongly
dipolar superfluids when the dipole polarisation is (anti-)parallel to the
initial vorticity of each vortex. The case when $\mathbf{B}\parallel\hat{x}$
is a little more ambiguous, though, as the evident drift to lower
$k_{\mathrm{c}}$ for higher $\varepsilon_{\mathrm{dd}}$ is not obeyed for all
of the choices of perturbation initial conditions. We note that each of the
initial conditions imply the same mode, $q=3$ becoming preferential for the
largest value of $\varepsilon_{\mathrm{dd}}$ considered in our work,
$\varepsilon_{\mathrm{dd}}=0.9$, but an explanation for the volatility of this
quantity with respect to the initial conditions for this particular dipole
alignment is not readily apparent.

Having identified the maximally unstable Kelvin mode, $k_{\mathrm{c}}$, in a
given dipolar regime, we have also studied the evolution of the corresponding
mode amplitude, $W(k_{\mathrm{c}},t)$, from the start of each dGPE simulation
till the reconnection induced by the Crow instability. After an early
transient arising from nonlinear interactions with the other Kelvin modes, we
find that $W(k_{\mathrm{c}},t)$ grows approximately exponentially in time,
i.e. $W(k_{\mathrm{c}},t)\sim\exp(\sigma t)$ with $\sigma$ the growth rate.
This is consistent with a linear dynamical instability of this mode arising
from the Crow mechanism, which is evident in Fig. 5 (a) where a plot of
$\log_{10}\,W(k_{\mathrm{c}},t)$ is approximately linear as a function of $t$
after sufficient time has passed. Here, $\varepsilon_{\mathrm{dd}}=0.9$ and
$\mathbf{B}\parallel\\{\hat{x},\hat{y},\hat{z}\\}$. Figure 5 (a) also
illustrates that the time taken till the first reconnection varies
considerably depending on the dipole polarisation, though it is known from
prior investigations that the velocities of vortex pairs are themselves
dipole-dependent [36].

Figure 5: (a) $\log_{10}\,W(k_{\mathrm{c}},t)$ as a function of $t$, with
$k_{\mathrm{c}}$ the maximally unstable mode wavenumber and
$\varepsilon_{\mathrm{dd}}=0.9$. The initial conditions correspond to those in
Figs. 1 and 2. (b) The ensemble-average of $\sigma$, the exponential growth
rate of $W(k_{\mathrm{c}},t)$, as a function of $\varepsilon_{\mathrm{dd}}$.
In both plots, $\mathbf{B}\parallel\\{\hat{x},\hat{y},\hat{z}\\}$.

To excise the transient, the exponential growth rate $\sigma$ has been
computed through linear regression fitting of $\log\,W(k_{\mathrm{c}},t)$ to
$t$ for times after the its growth is exponential, which for the sake of
consistency we take to be the interval
$t\in[t_{\mathrm{r}}/2,t_{\mathrm{c}}]$. The ensemble average of $\sigma$ over
the set of four Kelvin wave initial conditions is plotted as a function of
$\varepsilon_{\mathrm{dd}}$ as distinct curves for
$\mathbf{B}\parallel\\{\hat{x},\hat{y},\hat{z}\\}$ in Fig. 5 (b). In the
nondipolar limit, we can compare our computed value of the growth rate,
$\sigma\approx 0.0118$, to predictions of $\sigma$ obtained in the literature
through linear stability analysis of the corresponding Bogoliubov-de Gennes
(BdG) equations [15, 22]:

$\sigma(k_{\mathrm{c}})^{2}\sim\frac{k_{\mathrm{c}}^{2}}{d^{2}}\left[\ln(\sqrt{2}d)+0.38\right].$
(9)

This yields the estimation $\sigma\approx 0.04$; while the two predictions are
of the same order of magnitude, the discrepancy between the two may be due to
the nonlinear growth of $W(k_{\mathrm{c}},t)$ evident in Fig. 5 (a). For
finite values of $\varepsilon_{\mathrm{dd}}$, our results show that $\sigma$
is smaller when $\mathbf{B}\parallel\hat{x}$ than for the other two
polarisations regardless of $\varepsilon_{\mathrm{dd}}$ and that as
$\varepsilon_{\mathrm{dd}}\rightarrow 1$, a hierarchy such that
$\sigma(\hat{x})<\sigma(\hat{z})<\sigma(\hat{y})$ manifests itself. Now, given
that the initial mean separation of the vortices is identical for all of the
simulations we have conducted, one would expect that longer reconnection times
are consistent with smaller values of $\sigma$ and the hierarchy of $\sigma$
in Fig. 5 (b) is consistent with that of the ensemble-averaged reconnection
times,
$t_{\mathrm{r}}(\hat{x})>t_{\mathrm{r}}(\hat{z})>t_{\mathrm{r}}(\hat{y})$,
seen in Fig. 2 in the regime $\varepsilon_{\mathrm{dd}}=0.9$.

From our spectral analysis of the vortex lines during their evolution, it is
clear that there is a significant dependence on the polarisation of the dipole
moments relative to the initial vortex alignment and the DDI interaction
strength. The dependence of the maximally unstable mode $k_{\mathrm{c}}$ on
the characteristics of the DDI, in particular, suggests that local geometric
properties of the vortex line are sensitive to these parameters. One such
quantity is the curvature of the vortices, which we now proceed to as a
function of time for each of the simulations conducted. Parametrising a given
vortex line as $\boldsymbol{\gamma}(s)=\gamma_{i}(s)\hat{e}_{i}$, with
$\gamma_{z}(s)\equiv s$, its curvature is defined through this parametrisation
as [67]

$\kappa(s)=\frac{|\boldsymbol{\gamma}^{\prime}(s)\times\boldsymbol{\gamma}^{\prime\prime}(s)|}{|\boldsymbol{\gamma}^{\prime}(s)|^{3}}.$
(10)

In Eq. (10), we compute the derivatives directly from the mode amplitudes
$\\{W_{n}(k_{z},t)\\}$ by utilising the standard properties of the discrete
Fourier transform and by applying a simple low-pass filter such that only the
modes $L_{z}|k_{z}|/(2\pi)\leq 100$ contribute to the curvature 333Through
trial and error we have found that such a low-pass filter is necessary to
avoid aliasing effects, as well as noise from the vortex detection procedure,
that adversely affect the computation of higher-order derivatives..

Figure 6: The mean curvature over the length of the two vortices ensemble-
averaged over the set of vortex initial conditions, $\langle\kappa\rangle$, at
the time $t=t_{\mathrm{c}}$. The magnetic dipole orientation is parallel to
$\hat{x}$ in (a), $\hat{y}$ in (b), and $\hat{z}$ in (c).

Let us focus on the vortex profiles when $t=t_{\mathrm{c}}$ and compare the
curvature of the two vortices across the various regimes of
$\varepsilon_{\mathrm{dd}}$ and $\mathbf{B}$. Finding the mean curvatures of
both vortices over their respective line lengths and averaging the two
resulting quantities yields the quantity
$\langle\kappa(t_{\mathrm{c}})\rangle$, which is plotted in Fig. 6 as a
function of $\varepsilon_{\mathrm{dd}}$. Here, $\mathbf{B}$ is assumed to be
parallel to the $x$-, $y$\- and $z$-axes in (a), (b) and (c), respectively,
and for each choice of $\\{\mathbf{B},\,\varepsilon_{\mathrm{dd}}\\}$ we have
performed an ensemble average over the $4$ initial vortex perturbation
profiles. Figure 6 illustrates clearly the following salient characteristics
of the vortex line profiles arising from the Crow instability. While
fluctuations in the overall behaviour of the vortex lines occur due to the
specific choice of initial conditions, it is evident in Fig. 6 (a) that
$\langle\kappa(t_{\mathrm{c}})\rangle$ is largely independent of
$\varepsilon_{\mathrm{dd}}$ when the dipole moments are polarised along the
$x$-axis, the translation axis of the vortices, when compared to the $y$\- and
$z$-axes. As for dipole polarisations parallel to the $y$-axis, Fig. 6 (b)
suggests that $\langle\kappa(t_{\mathrm{c}})\rangle$ generally increases with
$\varepsilon_{\mathrm{dd}}$ and that the values it attains are comparable in
magnitude to the corresponding values for polarisations along the $x$-axis.

These observations are in sharp contrast to the results presented in Fig. 6
(c), where the mean vortex curvature at $t=t_{\mathrm{c}}$ is seen to be
strongly dependent on $\varepsilon_{\mathrm{dd}}$ when the dipole moments are
polarised along the $z$-axis, which is (anti-)parallel to the mean vorticity
of each axis. As $\varepsilon_{\mathrm{dd}}$ is increased, the curvature of
the vortex is suppressed rather than enhanced. This has been attributed in
earlier studies of Kelvin waves perturbing single, isolated vortices to the
effects of magnetostriction, where there exists an effective interaction
between the virtual dipole moments, induced in the vortex line, that are
polarised antiparallel to the real dipole moments in the bulk superfluid [51,
68, 52]. When $\mathbf{B}\parallel\hat{z}$, viz. (anti-)parallel to the mean
orientation of the vortex line, the effective dipolar interaction energy
arising from the mutual interaction of the virtual dipole moments is minimised
when the dipole moments are aligned along this mean orientation axis. Thus, a
suppression of the curvature of the vortex arises since a larger curvature
would effectively cause the configuration of the vortex’s virtual dipole
moments to become misaligned with the magnetic field and thus increase the
dipolar interaction energy. This also results in the wavenumber of the
preferentially excited Kelvin mode, $k_{\mathrm{c}}$, being smaller when
$\mathbf{B}\parallel\hat{z}$ than for polarisations along the other principal
axes as seen in Fig. 4. These effects are naturally more pronounced for larger
values of $\varepsilon_{\mathrm{dd}}$, which leads to smaller curvatures and
thus also the lower values of $k_{\mathrm{c}}$ seen in Figs. 4 (c).

We also note that magnetostriction arising from the virtual dipole moments in
the vortex cores serves to explain the salient features of the vortices when
$\mathbf{B}\parallel\hat{x},\,\hat{y}$ as well. When the dipole polarisation
are polarised along the $y$-axis, the energetic preference of the vortices’
virtual dipole moments is to align along this axis and such vortex curvature
is stimulated along this axis. While this results in the monotonic behaviour
of $\langle\kappa_{\mathrm{c}}\rangle$ in Fig. 6 (b), it is also responsible
for the reconnections occurring on a faster timescale at fixed
$\varepsilon_{\mathrm{dd}}$ than for the other polarisation orientations, as
observed in Fig. 2 and, as we have argued, also explains why the exponential
mode growth factor $\sigma$ is larger when $\mathbf{B}\parallel\hat{y}$ than
the other polarisations. Similarly, when the dipole polarisation is parallel
to the $x$-axis the contributions to the overall curvature arising from
curvature orthogonal to this axis are suppressed, such that the majority of
the curvature arises from curvature along the propagation axis of the vortex
pair. While this results in the strongly transversely perturbed vortex
profiles in the $x$-$z$ plane as is seen in Fig. 2 (a), curvature along the
$y$-axis is greatly inhibited and the reconnection is suppressed. Thus, the
exponential growth of the dominant Kelvin mode is suppressed by the DDI when
$\mathbf{B}\parallel\hat{x}$, resulting in smaller values of the mode growth
factor $\sigma$ and correspondingly longer reconnection times.

##  V Conclusion

Through studying the evolution of a pair of quantum vortices in a dipolar
superfluid with an initial transverse, symmetric perturbation applied to each
vortex, we have found that the quantum analogue of the Crow instability
behaves in vastly different ways depending on the magnetic dipole polarisation
and the relative strength of the DDI. When the dipole polarisation,
$\mathbf{B}$, is (anti-)parallel to the initial mean vorticity of each vortex,
the vortices’ curvature is strongly suppressed due to the influence of the
interaction between virtual dipole moments inside the vortex core. This
results in Kelvin waves of decreasing wavenumber contributing most strongly to
the Crow instability as the relative dipolar interaction strength,
$\varepsilon_{\mathrm{dd}}$, increases. This magnetostrictive effect has the
converse effect when the dipole polarisation is orthogonal to the initial
vorticities, with the curvature of each vortex as well as the growth of Kelvin
modes of higher wavenumber being enhanced by the influence of the DDI. While
the Kelvin mode occupation spectrum for a magnetic dipole polarisation
parallel to the separation between the two vortices appears rather similar to
its nondipolar analogue, these modes induce a greater curvature in the
vortices as $\varepsilon_{\mathrm{dd}}$ increases and, correspondingly, a Crow
instability that develops more rapidly. However, for dipole polarisations
parallel to the binormal axis, the Kelvin modes become concentrated at one
specific higher wavenumber mode, $p=3$, in the
large-$\varepsilon_{\mathrm{dd}}$ limit and the magnetostrictive elongation of
the vortex lines along the binormal axis suppresses the Crow instability when
compared to the other two dipole polarisations considered here.

Our studies of the dependence of the Crow instability on the orientation of
the dipole moments suggest that other hydrodynamical instabilities might
exhibit directional dependence in the presence of a DDI. Notably, the snake
instability of dark solitons in three-dimensional dipolar condensates has been
shown to be suppressed when the dipole alignment is parallel to the solitonic
phase slip [69]. The corresponding phase slip due to the vortex line pair in
our system is parallel to the $x$-axis and our results demonstrate a
suppression of the Crow instability for dipole polarisations parallel to this
axis. Given the intrinsic correspondence of the snake instability of exhibited
by dark solitons and the Crow instability of antiparallel vortex lines [15],
this is likely not a coincidence. This suppression of the snake instability
was demonstrated through linear stability analysis applied to the BdG spectra
of a three-dimensional dark soliton [69]; a similar procedure was conducted in
Ref. [22] to characterise the linear stability of antiparallel vortex lines in
a nondipolar BEC against the transverse perturbations that induce the Crow
instability. An application of linear stability analysis to the corresponding
dipolar problem as studied in this article may thus prove to be illuminating.
We also note that Ref. [69] studied only dipole orientations parallel to the
phase slip for dark solitons in three-dimensional regimes, and that our
analysis of the Crow instability for distinct dipole orientations suggest that
it is prudent to also examine the snake instability for dipole orientations in
the soliton plane.

We also note that our results bear implications for aspects of quantum vortex
dynamics in dipolar superfluids that, to the best of our knowledge, have not
yet been investigated thoroughly. For instance, in Sec. IV we have posited a
relationship between the exponential growth factor of the dominant Kelvin
mode, $\sigma$, and the time taken for the vortices to undergo their first
reconnection, $t_{\mathrm{r}}$. While the values of $\sigma$, ensemble-
averaged over the set of initial conditions for the vortex perturbation
profiles, exhibited a clear dependence on $\mathbf{B}$ and
$\varepsilon_{\mathrm{dd}}$, it would be desirable to investigate such scaling
behaviour in a setting that is independent of the randomness of the initial
conditions present in our system. In a dipolar superfluid, it would be of
considerable interest to elucidate the relationship between the parameters of
the DDI and the temporal scaling of quantities such as the inter-vortex
separation, the angle between the normal axes of the vortex lines, and global
properties such as the vortices’ curvature and torsion, and establish whether
or not they exhibit deviations from the familiar nondipolar paradigm [67, 70].
Furthermore,in the nondipolar limit a considerable degree of insight may be
gleaned into the dynamics of quantum vortex lines by assuming that the
superfluid flow is purely compressible, which allows the use of the Biot-
Savart law and the local induction approximation to model vortices before
reconnective processes occur [71, 72]. In this article, a strong influence has
been demonstrated of the magnetic dipole polarisation on the bending of vortex
line pairs, while the dynamics of quasi-two-dimensional point-vortices and
three-dimensional straight vortex lines have been shown to exhibit dipole-
mediated deviations from the predictions of the point-vortex model, the two-
dimensional limit of the Biot-Savart law. We also note that the Kelvin wave
dispersion relation for a single vortex line is modified in the presence of
the DDI [51, 52]. Given these pre-existing and new examples, an extension of
the Biot-Savart law [71, 72] to account for the dipolar interaction between
the virtual dipole moments in the vortex line cores would provide an intuitive
understanding of how these disparate results can be generalised for a larger
number of dipolar vortices. Furthermore, we also note that while this
investigation assumed a value of $\varepsilon_{\mathrm{dd}}$ less than $1$,
such that the dBEC does not exhibit the supersolid phase [43, 44, 45], the
interaction of supersolid quantum droplets and vortices produces complicated
dynamics that are inaccessible in the superfluid phase. It is feasible to
suggest that this might result in Kelvin waves and subsequently the Crow
instability in pairs of antiparallel vortices [73].

###### Acknowledgements.

This work was funded by Grant No. RPG-2021-108 from the Leverhulme Trust. S.
B. P. thanks Luca Galantucci for providing the pseudovorticity-based vortex
detection program used herein to extract the vortex line profiles. This
research made use of the Rocket High Performance Computing service at
Newcastle University.

## References

  * Crow [1970] S. C. Crow, Stability Theory for a Pair of Trailing Vortices, AIAA Journal 8, 2172 (1970).
  * Pitaevskii and Stringari [2016] L. Pitaevskii and S. Stringari,  _Bose-Einstein Condensation and Superfluidity_ , 1st ed., International Series of Monographs on Physics No. 164 (Clarendon Press, 2016).
  * Fetter [2009] A. L. Fetter, Rotating Trapped Bose-Einstein Condensates, Reviews of Modern Physics 81, 647 (2009).
  * Koplik and Levine [1993] J. Koplik and H. Levine, Vortex Reconnection in Superfluid Helium, Physical Review Letters 71, 1375 (1993).
  * Zuccher  _et al._ [2012] S. Zuccher, M. Caliari, A. W. Baggaley, and C. F. Barenghi, Quantum Vortex Reconnections, Physics of Fluids 24, 125108 (2012).
  * Ho [2001] T.-L. Ho, Bose-Einstein Condensates with Large Number of Vortices, Physical Review Letters 87, 060403 (2001).
  * Cooper  _et al._ [2001] N. R. Cooper, N. K. Wilkin, and J. M. F. Gunn, Quantum Phases of Vortices in Rotating Bose-Einstein Condensates, Physical Review Letters 87, 120405 (2001).
  * Baym [2003] G. Baym, Tkachenko Modes of Vortex Lattices in Rapidly Rotating Bose-Einstein Condensates, Physical Review Letters 91, 110402 (2003).
  * Watanabe  _et al._ [2004] G. Watanabe, G. Baym, and C. J. Pethick, Landau Levels and the Thomas-Fermi Structure of Rapidly Rotating Bose-Einstein Condensates, Physical Review Letters 93, 190401 (2004).
  * Henn  _et al._ [2009] E. A. L. Henn, J. A. Seman, G. Roati, K. M. F. Magalhães, and V. S. Bagnato, Emergence of Turbulence in an Oscillating Bose-Einstein Condensate, Physical Review Letters 103, 045301 (2009).
  * Tsatos  _et al._ [2016] M. C. Tsatos, P. E. S. Tavares, A. Cidrim, A. R. Fritsch, M. A. Caracanhas, F. E. A. dos Santos, C. F. Barenghi, and V. S. Bagnato, Quantum Turbulence in Trapped Atomic Bose-Einstein Condensates, Physics Reports 622, 1 (2016).
  * Navon  _et al._ [2016] N. Navon, A. L. Gaunt, R. P. Smith, and Z. Hadzibabic, Emergence of a Turbulent Cascade in a Quantum Gas, Nature 539, 72 (2016).
  * Madeira  _et al._ [2020] L. Madeira, M. A. Caracanhas, F. E. A. dos Santos, and V. S. Bagnato, Quantum Turbulence in Quantum Gases, Annual Review of Condensed Matter Physics 11, 37 (2020).
  * Barenghi  _et al._ [2023] C. F. Barenghi, H. A. J. Middleton-Spencer, L. Galantucci, and N. G. Parker, Types of Quantum Turbulence, AVS Quantum Science 5, 025601 (2023).
  * Kuznetsov and Rasmussen [1995] E. A. Kuznetsov and J. J. Rasmussen, Instability of Two-Dimensional Solitons and Vortices in Defocusing Media, Physical Review E 51, 4479 (1995).
  * Baggaley and Parker [2018] A. W. Baggaley and N. G. Parker, Kelvin-Helmholtz Instability in a Single-Component Atomic Superfluid, Physical Review A 97, 053608 (2018).
  * Giacomelli and Carusotto [2023] L. Giacomelli and I. Carusotto, Interplay of Kelvin-Helmholtz and Superradiant Instabilities of an Array of Quantized Vortices in a Two-Dimensional Bose-Einstein Condensate, SciPost Physics 14, 025 (2023).
  * Hernández-Rajkov  _et al._ [2024] D. Hernández-Rajkov, N. Grani, F. Scazza, G. Del Pace, W. J. Kwon, M. Inguscio, K. Xhani, C. Fort, M. Modugno, F. Marino, and G. Roati, Connecting Shear Flow and Vortex Array Instabilities in Annular Atomic Superfluids, Nature Physics 20, 939 (2024).
  * Sasaki  _et al._ [2009] K. Sasaki, N. Suzuki, D. Akamatsu, and H. Saito, Rayleigh-Taylor Instability and Mushroom-Pattern Formation in a Two-Component Bose-Einstein Condensate, Physical Review A 80, 063611 (2009).
  * Gautam and Angom [2010] S. Gautam and D. Angom, Rayleigh-Taylor Instability in Binary Condensates, Physical Review A 81, 053616 (2010).
  * Bezett  _et al._ [2010] A. Bezett, V. Bychkov, E. Lundh, D. Kobyakov, and M. Marklund, Magnetic Richtmyer-Meshkov Instability in a Two-Component Bose-Einstein Condensate, Physical Review A 82, 043608 (2010).
  * Berloff and Roberts [2001] N. G. Berloff and P. H. Roberts, Motion in a Bose Condensate: IX. Crow Instability of Antiparallel Vortex Pairs, Journal of Physics A: Mathematical and Theoretical 34, 10057 (2001).
  * Simula [2011] T. P. Simula, Crow Instability in Trapped Bose-Einstein Condensates, Physical Review A 84, 021603(R) (2011).
  * Gautam [2013] S. Gautam, Crow Instability in Unitary Fermi Gas, Modern Physics Letters B 27, 1350097 (2013).
  * Villois  _et al._ [2017] A. Villois, D. Proment, and G. Krstulovic, Universal and Nonuniversal Aspects of Vortex Reconnections in Superfluids, Physical Review Fluids 2, 044701 (2017).
  * Lahaye  _et al._ [2009] T. Lahaye, C. Menotti, L. Santos, M. Lewenstein, and T. Pfau, The Physics of Dipolar Bosonic Quantum Gases, Reports on Progress in Physics 72, 126401 (2009).
  * Martin  _et al._ [2017] A. M. Martin, N. G. Marchant, D. H. J. O’Dell, and N. G. Parker, Vortices and Vortex Lattices in Quantum Ferrofluids, Journal of Physics: Condensed Matter 29, 103004 (2017).
  * Chomaz  _et al._ [2023] L. Chomaz, I. Ferrier-Barbut, F. Ferlaino, B. Laburthe-Tolra, B. L. Lev, and T. Pfau, Dipolar Physics: A Review of Experiments with Magnetic Quantum Gases, Reports on Progress in Physics 86, 026401 (2023).
  * Eberlein  _et al._ [2005] C. Eberlein, S. Giovanazzi, and D. H. J. O’Dell, Exact Solution of the Thomas-Fermi Equation for a Trapped Bose-Einstein Condensate with Dipole-Dipole Interactions, Physical Review A 71, 033618 (2005).
  * Stuhler  _et al._ [2005] J. Stuhler, A. Griesmaier, T. Koch, M. Fattori, T. Pfau, S. Giovanazzi, P. Pedri, and L. Santos, Observation of Dipole-Dipole Interaction in a Degenerate Quantum Gas, Physical Review Letters 95, 150406 (2005).
  * Ticknor  _et al._ [2011] C. Ticknor, R. M. Wilson, and J. L. Bohn, Anisotropic Superfluidity in a Dipolar Bose Gas, Physical Review Letters 106, 065301 (2011).
  * Wenzel  _et al._ [2018] M. Wenzel, F. Böttcher, J.-N. Schmidt, M. Eisenmann, T. Langen, T. Pfau, and I. Ferrier-Barbut, Anisotropic Superfluid Behaviour of a Dipolar Bose-Einstein Condensate, Physical Review Letters 121, 030401 (2018).
  * Abad  _et al._ [2009] M. Abad, M. Guilleumas, R. Mayol, M. Pi, and D. M. Jezek, Vortices in Bose-Einstein Condensates with Dominant Dipole-Dipole Interactions, Physical Review A 79, 063622 (2009).
  * Mulkerin  _et al._ [2013] B. C. Mulkerin, R. M. W. van Bijnen, D. H. J. O’Dell, A. M. Martin, and N. G. Parker, Anisotropic and Long-Range Vortex Interactions in Two-Dimensional Dipolar Bose Gases, Physical Review Letters 111, 170402 (2013).
  * Bland  _et al._ [2023] T. Bland, G. Lamporesi, M. J. Mark, and F. Ferlaino, Vortices in Dipolar Bose-Einstein Condensates, Comptes Rendus Physique 24, 1 (2023).
  * Prasad  _et al._ [2024] S. B. Prasad, N. G. Parker, and A. W. Baggaley, Vortex-Pair Dynamics in Three-Dimensional Homogeneous Dipolar Superfluids, Physical Review A 109, 063323 (2024).
  * Cooper  _et al._ [2005] N. R. Cooper, E. H. Rezayi, and S. H. Simon, Vortex Lattices in Rotating Atomic Bose Gases with Dipolar Interactions, Physical Review Letters 95, 200402 (2005).
  * Zhang and Zhai [2005] J. Zhang and H. Zhai, Vortex Lattices in Planar Bose-Einstein Condensates with Dipolar Interactions, Physical Review Letters 95, 200403 (2005).
  * Cai  _et al._ [2018] Y. Cai, Y. Yuan, M. Rosenkranz, H. Pu, and W. Bao, Vortex Patterns and the Critical Rotation Frequency in Rotating Dipolar Bose-Einstein Condensates, Physical Review A 98, 023610 (2018).
  * Bland  _et al._ [2018] T. Bland, G. W. Stagg, L. Galantucci, A. W. Baggaley, and N. G. Parker, Quantum Ferrofluid Turbulence, Physical Review Letters 121, 174501 (2018).
  * Wilson  _et al._ [2012] R. M. Wilson, C. Ticknor, J. L. Bohn, and E. Timmermans, Roton Immiscibility in a Two-Component Dipolar Bose Gas, Physical Review A 86, 033606 (2012).
  * Hertkorn  _et al._ [2021] J. Hertkorn, J.-N. Schmidt, M. Guo, F. Böttcher, K. S. H. Ng, S. D. Graham, P. Uerlings, T. Langen, M. Zwierlein, and T. Pfau, Pattern Formation in Quantum Ferrofluids: From Supersolids to Superglasses, Physical Review Research 3, 033125 (2021).
  * Böttcher  _et al._ [2019] F. Böttcher, J.-N. Schmidt, M. Wenzel, J. Hertkorn, M. Guo, T. Langen, and T. Pfau, Transient Supersolid Properties in an Array of Dipolar Quantum Droplets, Physical Review X 9, 011051 (2019).
  * Chomaz  _et al._ [2019] L. Chomaz, D. Petter, P. Ilzhöfer, G. Natale, A. Trautmann, C. Politi, G. Durastante, R. M. W. van Bijnen, A. Patscheider, M. Sohmen, M. J. Mark, and F. Ferlaino, Long-Lived and Transient Supersolid Behaviors in Dipolar Quantum Gases, Physical Review X 9, 021012 (2019).
  * Tanzi  _et al._ [2019] L. Tanzi, E. Lucioni, F. Famà, J. Catani, A. Fioretti, C. Gabbanini, R. N. Bisset, L. Santos, and G. Modugno, Observation of a Dipolar Quantum Gas with Metastable Supersolid Properties, Physical Review Letters 122, 130405 (2019).
  * Klaus  _et al._ [2022] L. Klaus, T. Bland, E. Poli, C. Politi, G. Lamporesi, E. Casotti, R. N. Bisset, M. J. Mark, and F. Ferlaino, Observation of Vortices and Vortex Stripes in a Dipolar Condensate, Nature Physics 18, 1453 (2022).
  * Casotti  _et al._ [2024] E. Casotti, E. Poli, L. Klaus, A. Litvinov, C. Ulm, C. Politi, M. J. Mark, T. Bland, and F. Ferlaino, Observation of vortices in a dipolar supersolid, (2024), arXiv:2403.18510 [cond-mat.quant-gas] .
  * Gautam [2014] S. Gautam, Dynamics of the Corotating Vortices in Dipolar Bose–Einstein Condensates in the Presence of Dissipation, Journal of Physics B: Atomic, Molecular and Optical Physics 47, 165301 (2014).
  * Zhao [2021] Q. Zhao, Effects of Dipole–Dipole Interaction on Vortex Motion in Bose–Einstein Condensates, Journal of Low Temperature Physics 204, 1 (2021).
  * Sabari  _et al._ [2024] S. Sabari, R. Kishor Kumar, and L. Tomio, Vortex Dynamics and Turbulence in Dipolar Bose-Einstein Condensates, Physical Review A 109, 023313 (2024).
  * Klawunn  _et al._ [2008] M. Klawunn, R. Nath, P. Pedri, and L. Santos, Transverse Instability of Straight Vortex Lines in Dipolar Bose-Einstein Condensates, Physical Review Letters 100, 240403 (2008).
  * Lee  _et al._ [2018] A.-C. Lee, D. Baillie, R. N. Bisset, and P. B. Blakie, Excitations of a Vortex Line in an Elongated Dipolar Condensate, Physical Review A 98, 063620 (2018).
  * Schützhold  _et al._ [2006] R. Schützhold, M. Uhlmann, Y. Xu, and U. R. Fischer, Mean-Field Expansion in Bose–Einstein Condensates with Finite-Range Interactions, International Journal of Modern Physics B 20, 3555 (2006).
  * Lima and Pelster [2012] A. R. P. Lima and A. Pelster, Beyond Mean-Field Low-Lying Excitations of Dipolar Gases, Physical Review A 86, 063609 (2012).
  * Bisset  _et al._ [2016] R. N. Bisset, R. M. Wilson, D. Baillie, and P. B. Blakie, Ground-State Phase Diagram of a Dipolar Condensate with Quantum Fluctuations, Physical Review A 94, 033619 (2016).
  * Schmitt  _et al._ [2016] M. Schmitt, M. Wenzel, F. Böttcher, I. Ferrier-Barbut, and T. Pfau, Self-Bound Droplets of a Dilute Magnetic Quantum Liquid, Nature 539, 259 (2016).
  * Chomaz  _et al._ [2016] L. Chomaz, S. Baier, D. Petter, M. J. Mark, F. Wächtler, L. Santos, and F. Ferlaino, Quantum-Fluctuation-Driven Crossover from a Dilute Bose-Einstein Condensate to a Macrodroplet in a Dipolar Quantum Fluid, Physical Review X 6, 041039 (2016).
  * Barenghi and Parker [2016] C. F. Barenghi and N. G. Parker,  _A Primer on Quantum Fluids_ , Springer Briefs in Physics (Springer, 2016).
  * Bao and Cai [2013] W. Bao and Y. Cai, Mathematical Theory and Numerical Methods for Bose-Einstein Condensation, Kinetic and Related Models 6, 1 (2013).
  * Villois  _et al._ [2016] A. Villois, G. Krstulovic, D. Proment, and H. Salman, A Vortex Filament Tracking Method for the Gross–Pitaevskii Model of a Superfluid, Journal of Physics A: Mathematical and Theoretical 49, 415502 (2016).
  * Serafini  _et al._ [2017] S. Serafini, L. Galantucci, E. Iseni, T. Bienaimé, R. N. Bisset, C. F. Barenghi, F. Dalfovo, G. Lamporesi, and G. Ferrari, Vortex Reconnections and Rebounds in Trapped Atomic Bose-Einstein Condensates, Physical Review X 7, 021031 (2017).
  * Hossain  _et al._ [2022] K. Hossain, K. Kobuszewski, M. Forbes, P. Magierski, K. Sekizawa, and G. Wlazłowski, Rotating Quantum Turbulence in the Unitary Fermi Gas, Physical Review A 105, 013304 (2022).
  * Middleton-Spencer  _et al._ [2023] H. A. J. Middleton-Spencer, A. D. G. Orozco, L. Galantucci, M. Moreno, N. G. Parker, L. A. Machado, V. S. Bagnato, and C. F. Barenghi, Strong Quantum Turbulence in Bose-Einstein Condensates, Physical Review Research 5, 043081 (2023).
  * Weiss and McWilliams [1991] J. B. Weiss and J. C. McWilliams, Nonergodicity of Point Vortices, Physics of Fluids A: Fluid Dynamics 3, 835 (1991).
  * Billam  _et al._ [2014] T. P. Billam, M. T. Reeves, B. P. Anderson, and A. S. Bradley, Onsager-Kraichnan Condensation in Decaying Two-Dimensional Quantum Turbulence, Physical Review Letters 112, 145301 (2014).
  * Griffin  _et al._ [2020] A. Griffin, V. Shukla, M.-E. Brachet, and S. Nazarenko, Magnus-Force model for Active Particles Trapped on Superfluid Vortices, Physical Review A 101, 053601 (2020).
  * Rorai  _et al._ [2016] C. Rorai, J. Skipper, R. M. Kerr, and K. R. Sreenivasan, Approach and Separation of Quantised Vortices with Balanced Cores, Journal of Fluid Mechanics 808, 641 (2016).
  * Klawunn and Santos [2009] M. Klawunn and L. Santos, Phase Transition from Straight into Twisted Vortex Lines in Dipolar Bose–Einstein Condensates, New Journal of Physics 11, 055612 (2009).
  * Nath  _et al._ [2008] R. Nath, P. Pedri, and L. Santos, Stability of Dark Solitons in Three Dimensional Dipolar Bose-Einstein Condensates, Physical Review Letters 101, 210402 (2008).
  * Galantucci  _et al._ [2019] L. Galantucci, A. W. Baggaley, N. G. Parker, and C. F. Barenghi, Crossover from Interaction to Driven Regimes in Quantum Vortex Reconnections, Proceedings of the National Academy of Sciences 116, 12204 (2019).
  * Schwarz [1985] K. W. Schwarz, Three-Dimensional Vortex Dynamics in Superfluid 4He: Line-Line and Line-Boundary Interactions, Physical Review B 31, 5782 (1985).
  * Schwarz [1988] K. W. Schwarz, Three-Dimensional Vortex Dynamics in Superfluid 4He: Homogeneous Superfluid Turbulence, Physical Review B 38, 2398 (1988).
  * Poli  _et al._ [2023] E. Poli, T. Bland, S. J. M. White, M. J. Mark, F. Ferlaino, S. Trabucco, and M. Mannarelli, Glitches in Rotating Supersolids, Physical Review Letters 131, 223401 (2023).