#  Coexistence of vortex arrays and surface capillary waves in spinning
prolate superfluid 4He nanodroplets

Martí Pi  Departament FQA, Facultat de Física, Universitat de Barcelona, Av.
Diagonal 645, 08028 Barcelona, Spain.  Institute of Nanoscience and
Nanotechnology (IN2UB), Universitat de Barcelona, Barcelona, Spain.  José
María Escartín  Catalan Institute of Nanoscience and Nanotechnology (ICN2),
CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain.  Francesco
Ancilotto  Dipartimento di Fisica e Astronomia “Galileo Galilei” and CNISM,
Università di Padova, via Marzolo 8, 35122 Padova, Italy.  CNR-IOM Democritos,
via Bonomea, 265 - 34136 Trieste, Italy.  Manuel Barranco  Departament FQA,
Facultat de Física, Universitat de Barcelona, Av. Diagonal 645, 08028
Barcelona, Spain.  Institute of Nanoscience and Nanotechnology (IN2UB),
Universitat de Barcelona, Barcelona, Spain.

###### Abstract

Within Density Functional Theory, we have studied the interplay between vortex
arrays and capillary waves in spinning prolate 4He droplets made of several
thousands of helium atoms. Surface capillary waves are ubiquitous in prolate
superfluid 4He droplets and, depending on the size and angular momentum of the
droplet, they may coexist with vortex arrays. We have found that the
equilibrium configuration of small prolate droplets is vortex-free, evolving
towards vortex-hosting as the droplet size increases. This result is in
agreement with a recent experiment [S. M. O. O’Connell et at., Phys. Rev.
Lett. 124, 215301 (2020)], where it has been disclosed that vortex arrays and
capillary waves coexist in the equilibrium configuration of very large drops.
Contrarily to viscous droplets executing rigid body rotation, the stability
phase diagram of spinning 4He droplets cannot be universally described in
terms of dimensionless angular momentum and angular velocity variables:
instead, the rotational properties of superfluid helium droplets display a
clear dependence on the droplet size and the number of vortices they host.

##  I Introduction

Superfluid 4He droplets produced in the expansion of a cold helium gas,Toe04
or by hydrodynamic instability of a liquid helium jet passing through the
nozzle of a molecular beam apparatus,Tan18  have been considered the ultimate
inert matrix for molecular spectroscopy,Leh98 ; Cal11  constituting also an
ideal testground to study superfluidity at the nanoscale.Sin89 ; Kri90 ; Gre98
; Kro01 ; Tan02 ; Bra13 ; Mat15  A series of experiments aiming at determining
the appearance of large spinning 4He droplets made of $10^{8}-10^{11}$
atomsGom14 ; Jon16 ; Ber17 ; Rup17 ; Lan18 ; Ges19 ; Oco20  has motivated a
renewed interest in aspects such as how vortices are distributed inside helium
droplets,Anc15 ; Anc18  how impurities are captured by the vortex lines they
host, and how impurities arrange inside vortex-hosting droplets.Gom12 ; Lat14
; Vol15 ; Cop17 ; Bla18 ; Cop19 ; Ern21

Two experimental techniques have been used to address large spinning 4He
droplets. The pioneering work of Ref. Gom14,  and other works carried out by
the same groupJon16 ; Ber17 ; Oco20  used coherent diffractive imaging of
x-rays from a free electron laser (FEL) and gave access to a model-independent
determination of the two-dimensional (2D) projection of the droplet density on
a plane perpendicular to the x-ray incident direction via iterative phase
retrieval algorithms; by doping He droplets with Xe atoms they were also able
to detect the presence of vortex arrays. In Refs. Rup17, ; Lan18, , He
droplets were irradiated with intense extreme ultraviolet pulses (XUV), and
the measurements of wide-angle diffraction patterns provided access to full
three-dimensional information. However, it was not possible to directly
determine the shapes of the droplets, and so the analysis was carried out
parameterizing the droplet density with a combination of two ellipsoidal caps
connected by a hyperboloidal centerpiece, and all their defining parameters
were determined by matching the experimental diffraction patterns with those
obtained from simulations.

These remarkable experiments have shown that most helium droplets are
spherical and only a few percent are deformed. In particular, the analysis
carried out by Langbehn et al.Lan18  on a sample consisting of a large number
of droplets (38,150) showed that 92.9% of them were spherical, 5.6% were
spheroidal (oblatenote ), and 1.5% were prolate. The latter population was
made of ellipsoidal (0.8%), pill-shaped (0.6%), and dumbbell-like (0.1%)
droplets, where the appearance was quantitatively determined by evaluating the
distance of the center of mass of the droplet to its surface along the
direction of the principal axes of inertia. The results obtained with both
experimental techniques were compared to calculations made for incompressible
viscous droplets only subject to surface tension and centrifugal forces;Bro80
; Hei06 ; But11 ; Bal15  it was concluded that they were in agreement.

The precise meaning of this agreement needs some clarification, as otherwise
one might be left with the impression that the rotational properties of
superfluid 4He and of classical rotating viscous fluids are very similar, and
this is not the case.Lam45 ; Guy15  The experiments have indeed shown that the
shapes of spinning superfluid 4He drops are the same as those found for
viscous drops subject to centrifugal and surface tension forces, and that some
relationships between ratios of the mentioned distances are in agreement with
the classical results,Bal15  see, e.g., Fig. 3 of Ref. Lan18,  and Fig. 1
below. However, to ascertain whether the sequence of shapes is the same, which
is what determines the equilibrium phase diagram, it is necessary to
experimentally determine the angular momentum for a large enough sample of
droplets; this has not been possible yet. Only very recently, the angular
momentum and shape of a few He drops have been simultaneously determined.Oco20

At first glance, the mere fact that oblate 4He spinning drops have been
identifiedGom14 ; Lan18 ; Oco20  is surprising and unexpected. Large helium
drops are produced in the normal phase and may acquire angular momentum during
the passage of the fluid through the nozzle of the molecular beam apparatus.
The drops cool down to a temperature of 0.37 KToe04  and are thus superfluid
when they are probed. The natural question is how a spinning superfluid drop
may acquire an oblate (axisymmetric) shape as this is quantum mechanically
forbidden. In fact the equations characterizing the macroscopic behaviour of a
superfluid at zero temperature have the form of irrotational hydrodynamics,
from which the moment of inertia can be calculated asCop17 ; Boh75 ; Pit16
$\Theta_{\rm irr}=\varepsilon^{2}\,\Theta_{\rm rig}$, where

$\varepsilon=\frac{\langle y^{2}-x^{2}\rangle}{\langle y^{2}+x^{2}\rangle}$
(1)

and $\Theta_{\rm rig}=Nm\langle x^{2}+y^{2}\rangle$ is the rigid moment of
inertia, $N$ being the number of atoms in the droplet and $m$ the atomic mass,
showing that in a superfluid the value of the moment of inertia is smaller
than the rigid value. In particular, for axisymmetric systems (i.e. $\langle
x^{2}\rangle=\langle y^{2}\rangle$) the above relation gives for the angular
momentum of the superfluid $\langle L_{z}\rangle=\Theta_{\rm irr}\,\omega=0$.
Notice that these results are not restricted to the perturbative regime of
small $\omega$.Boh75  These expressions show that vortex-free oblate
axisymmetric configurations cannot spin, whereas prolate (non axisymmetric)
configurations can, and the resulting angular momentum $L_{\rm
cap}=\Theta_{\rm irr}\,\omega$ is associated with the so-called capillary
waves (see the discussion in the following).

Thus, the only likely explanation for the experimental observation of spinning
oblate 4He droplets is that these drops contain a vortex array which can store
most of the angular momentum acquired during the passage through the
experimental apparatus –some is taken away by the atoms emitted during the
cooling process. It has indeed been shownAnc15 ; Anc18  that the presence of a
large enough vortex array in the 4He droplets confers to them a globally
oblate appearance,note  similarly to the case of rotating viscous droplets.
Vortex arrays have been experimentally detected only in a few cases,Gom14 ;
Jon16 ; Ber17 ; Oco20  as they escape detection unless drops are doped and the
presence of vortices is established by the appearance of Bragg patterns from
the impurities that fill the vortex cores.

The existence of helium drops that follow the sequence of prolate shapes
characteristic of viscous drops under the effect of rotation is also worth
discussing. In this case, axial symmetry is spontaneously broken ($\langle
x^{2}\rangle\neq\langle y^{2}\rangle$), and the droplet can store a finite
amount of angular momentum in the form of capillary waves, even in the absence
of vortices. In general, however, capillary waves and vortices may coexist in
spinning prolate droplets, producing two quite different irrotational velocity
flows compatible with the superfluidity requirement: one associated to
quantized vortices, and the other to surface capillary waves.Boh75 ; Don91 ;
Pit16 ; Sei94  Remarkably, in the stability phase diagram, spinning prolate
superfluid drops whose angular momentum is exclusively stored in capillary
waves lie on a lineAnc18  quite distinct from the classical line,Bro80  as
discussed in the following.

There is another unexpected difference between a rotational viscous fluid and
an irrotational superfluid with finite angular momentum. When a spherical
viscous droplet is set into rotation, as a consequence of centrifugal forces
it flattens out in the direction of the rotational axis: the shortest half-
axis of the droplet ($c_{z}$, as defined in the following) is thus aligned
with the rotational axis. However, this is not always the case for
irrotational flows: spinning vortex-free droplets with prolate shapes have
instead the intermediate half-axis ($b_{y}$, as defined in the following)
aligned with the rotational axis.Lam45  When vortices are present, in most
cases the shortest half-axis $c_{z}$ is again aligned with the rotational
axis, in agreement with the classical behavior. Consequently, if only the
shape of the droplet is experimentally identified, the rotational axis cannot
be determined unambiguously if no further information about the angular
momentum vector is available.

From the above discussion it appears that capillary waves and vortex arrays
can coexist in prolate superfluid drops. This has been demonstrated in a
recent combined experimental and Density Functional Theory (DFT) work.Oco20
The DFT analysis was restricted to deformable self-sustained superfluid 4He
cylinders which could host a fairly large vortex array due to the lower
dimensionality (2D) of the computational real space grid. This model was a
generalization of the rotating, elliptic cylindric vessel filled with 4He
studied in Ref. Fet74, .

In this work we complement the discussion made in Ref. Oco20,  by addressing
several prolate configurations corresponding to large (for DFT standards)
superfluid 4He droplets: we put the emphasis on the interplay between vortices
and capillary waves, we determine the equilibrium configuration for two
representative values of the angular momentum, and we discuss the similarities
and differences between superfluid (irrotational) and viscous (rotational)
behavior. Oblate droplets hosting a different number of vortices have been
thoroughly studied in Ref. Anc15, .

This paper is organized as follows: the DFT approach to liquid helium is
recalled in Sec. II; Sec. III presents the results and their discussion; and a
summary is given in Sec. IV. Multimedia information is presented in the
Supplemental Material.SM

##  II Theoretical Approach

Density Functional Theory in its static and time-dependent (TDDFT) versions
has proved to be a powerful method to address superfluid helium droplets.Bar06
; Anc17  Within DFT, the energy of the 4He droplet is written as a functional
of the atom density $\rho({\mathbf{r}})$ as

$E[\rho]=T[\rho]+E_{c}[\rho]=\frac{\hbar^{2}}{2m}\int
d{\mathbf{r}}|\nabla\Psi({\mathbf{r}})|^{2}+\int d{\mathbf{r}}\,{\cal
E}_{c}[\rho]$ (2)

where the first term is the kinetic energy, with
$\rho({\mathbf{r}})=|\Psi({\mathbf{r}})|^{2}$, and the functional ${\cal
E}_{c}$ contains the interaction term (in the Hartree approximation) and
additional terms which describe non-local correlation effects.Anc05

The droplet equilibrium configuration is obtained by solving the Euler-
Lagrange (EL) equation resulting from the functional variation of Eq. (2),

$\left\\{-\frac{\hbar^{2}}{2m}\nabla^{2}+\frac{\delta{\cal
E}_{c}}{\delta\rho}\right\\}\Psi({\mathbf{r}})\equiv{\cal
H}[\rho]\,\Psi({\mathbf{r}})=\mu\Psi({\mathbf{r}})\;,$ (3)

where $\mu$ is the 4He chemical potential corresponding to the number of He
atoms in the droplet, $N=\int d{\bf r}|\Psi({\bf r})|^{2}$.

To study spinning 4He droplets it is convenient to work in the fixed-droplet
frame of reference (corotating frame at angular velocity $\omega$), i.e., we
consider

$E^{\prime}[\rho]=E[\rho]-\hbar\omega\,\langle\hat{L}_{z}\rangle$ (4)

where $\hat{L}_{z}$ is the dimensionless angular momentum operator in the
$z$-direction; one looks for solutions of the EL equation resulting from the
functional variation of $E^{\prime}[\rho]$,

$\left\\{{\cal
H}[\rho]\,-\hbar\omega\hat{L}_{z}\right\\}\,\Psi(\mathbf{r})=\,\mu\,\Psi(\mathbf{r})\;.$
(5)

The above equation has been solved using the 4He-DFT-BCN-TLS computing
package,Pi17  see Refs. Anc17,  and dft-guide,  and references therein for
additional details. Briefly, we work in cartesian coordinates, with the
effective wave function $\Psi(\mathbf{r})$ defined at the nodes of a 3D grid
inside a calculation box large enough to accommodate the droplet in such a way
that the He density is sensibly zero on the box surface. Periodic boundary
conditions are imposed which allow to use the Fast Fourier Transform to carry
out the convolutions needed to obtain the DFT mean field ${\cal H}[\rho]$. The
differential operators in ${\cal H}[\rho]$ and $\hat{L}_{z}$ are approximated
by 13-point formulas and the typical space step is 0.8 Å. Working at fixed
angular momentum requires to iterate on the value of $\omega$; there are
efficient ways of adjusting iteratively $\omega$, e.g., the Augmented
Lagrangian method.Gar21

$\Lambda=0.80$ $L=4155$ $L_{\rm cap}=4014$ $N=1500$
---
$n_{v}$ |  $E$ (K) |  $E_{\rm kin}$ (K) |  $\omega$ (K/$\hbar$) |  $a_{x}$ (Å) | $b_{y}{\rm(\AA)}$ | $c_{z}{\rm(\AA)}$
0 | -7909.33 | 419.82 | 0.1145 | 42.38 | 15.93 | 18.93
1 | -7916.16 | 478.10 | 0.1269 | 39.75 | 18.53 | 18.07
2 | -7918.89 | 533.96 | 0.1349 | 36.50 | 20.71 | 19.32
3(T) | -7905.11 | 594.92 | 0.1431 | 32.76 | 24.13 | 19.56
4(C) | -7885.24 | 635.61 | 0.1315 | 28.69 | 28.39 | 20.32
Table 1:  Energetics and morphology of prolate 4He1500 droplets at
$\Lambda=0.80$ for $n_{v}=0-4$. $T$ indicates a triangular and $C$ a cross-
shaped vortex array. The entry in boldface corresponds to the globally stable
configuration.

Angular momentum can be stored in a superfluid 4He droplet in the form of
surface capillary waves and/or quantized vortices.Sei94 ; Anc18 ; Oco20  In
order to deposit angular momentum of both types in the droplet, we have used
the so-called “imprinting” procedure by starting the imaginary time
minimization from a very flexible guess for the effective wave function
$\Psi(\mathbf{r})$, namely a superposition of a quadrupolar capillary wave and
$n_{v}$ vortex lines parallel to the $z$ axis,

$\Psi_{0}(\mathbf{r})=\rho_{0}^{1/2}(\mathbf{r})\,e^{\imath\,\alpha
xy}\,\prod_{j=1}^{n_{v}}{(x-x_{j})+\imath(y-y_{j})\over\sqrt{(x-x_{j})^{2}+(y-y_{j})^{2}}}\;,$
(6)

which is then optimized by iteratively solving Eq. (5). Here,
$\rho_{0}(\mathbf{r})$ is an arbitrary, vortex-free droplet density, the
complex phase $e^{\imath\alpha xy}$ imprints a surface capillary wave with
quadrupolar symmetry around the $z$ axis,Cop17  and the product term imprints
a vortex array made of $n_{v}$ linear vortices,Anc17  where $(x_{j},y_{j})$ is
the initial position of the $j$-vortex core. The initial value of $\alpha$ and
the vortex core positions are guessed, and during the iterative solution of
Eq. (5) both the vortex core structure and positions, and the droplet shape
change to provide at convergence the lowest energy configuration.

$\Lambda=0.80$ $L=85317$ $L_{\rm cap}=82449$ $N=20{,}000$
---
$n_{v}$ |  $E$ (K) |  $E_{\rm kin}$ (K) |  $\omega$ (K/$\hbar$) |  $a_{x}$ (Å) | $b_{y}{\rm(\AA)}$ | $c_{z}{\rm(\AA)}$
0 | -127236.24 | 2319.50 | 0.03172 | 100.07 | 37.75 | 44.90
1 | -127320.36 | 2495.08 | 0.03422 | 96.22 | 41.65 | 44.75
2 | -127397.02 | 2654.06 | 0.03607 | 91.70 | 44.51 | 45.97
3(L) | -127432.28 | 2831.78 | 0.03733 | 87.11 | 47.88 | 44.73
4(C) | -127428.73 | 3021.48 | 0.03908 | 82.20 | 52.50 | 46.62
4∗(L) | -127405.10 | 3027.91 | 0.03745 | 82.85 | 51.37 | 46.43
5(P) | -127413.79 | 3214.34 | 0.04021 | 75.66 | 58.05 | 47.34
Table 2:  Energetics and morphology of prolate 4He droplets with $N=20{,}000$
at $\Lambda=0.80$ for $n_{v}=0-5$. $L$ indicates a linear, $C$ a cross-shaped,
and $P$ a pentagonal vortex array. The entry in boldface corresponds to the
globally stable configuration, and the asterisk identifies the highest energy
$n_{v}=4$ configuration.

Writing
$\Psi(\mathbf{r})\equiv\phi(\mathbf{r})\,\exp[i\,\cal{S}(\mathbf{r})]$, the
velocity field of the superfluid is

$\mathbf{v}(\mathbf{r})=\frac{\hbar}{m}{\rm
Im}\left\\{\frac{\nabla\Psi(\mathbf{r})}{\Psi(\mathbf{r})}\right\\}=\frac{\hbar}{m}\nabla\cal{S}(\mathbf{r})$
(7)

The velocity potential $\cal{S}(\mathbf{r})$ gets contributions from both
capillary waves and vortices in a complex way, and their contributions cannot
be rigorously disentangled —the analysis needs to be model-dependent.Oco20

$\Lambda=0.80$ $L=163899$ $L_{\rm cap}=158398$ $N=35{,}000$
---
$n_{v}$ |  $E$ (K) |  $E_{\rm kin}$ (K) |  $\omega$ (K/$\hbar$) |  $a_{x}$ (Å) |  $b_{y}$ (Å) |  $c_{z}$ (Å)
0 | -227393.50 | 3360.11 | 0.02402 | 120.49 | 45.42 | 54.21
1 | -227522.50 | 3581.02 | 0.02580 | 116.29 | 49.68 | 54.09
2 | -227629.67 | 3767.38 | 0.02697 | 111.55 | 52.80 | 55.29
3 | -227698.13 | 3986.17 | 0.02789 | 106.56 | 56.28 | 53.89
4(C) | -227721.40 | 4233.29 | 0.02927 | 101.38 | 61.12 | 56.03
4∗(L) | -227700.04 | 4233.98 | 0.02829 | 101.76 | 60.15 | 55.82
5(P) | -227723.95 | 4476.43 | 0.03009 | 95.16 | 66.16 | 56.86
6(Ct) | -227699.01 | 4732.32 | 0.03130 | 85.12 | 75.48 | 53.36
6∗ | -227697.29 | 4710.33 | 0.03012 | 87.06 | 73.60 | 57.59
Table 3:  Energetics and morphology of prolate 4He droplets with $N=35{,}000$
at $\Lambda=0.80$ for $n_{v}=0-6$. $L$ indicates a linear, $C$ a cross-shaped,
and $P$ a pentagonal vortex array. The 6($Ct$) configuration has a center
vortex, i.e., along the rotational axis. The entry in boldface corresponds to
the globally stable configuration, and the asterisk identifies the highest
energy configuration among those with the same $n_{v}$ value; in particular,
6∗ corresponds to a 6-vortex array configuration in which the six vortices are
at the vertices of a hexagon.

The existence of different prolate configurations for a 4HeN droplet spinning
at a given $L$, each of them characterized by a number of vortices $n_{v}$,
raises the question of which is the globally stable configuration and how does
it depend on the number of atoms in the droplet. To determine which is the
globally stable configuration among different ones, one has to compare their
total energy $E[\rho]$ including the rotational energy (Routhian).Bro80 ;
Hei06  At variance with the classical case, where a rotational energy term in
the rigid body approximation has to be added to the energy of the
droplet,Bro80 ; But11  in the DFT approach to superfluid 4He in the corotating
frame this is naturally accounted for through the velocity field embodied in
the phase of the effective wave function $\Psi({\mathbf{r}})$, see Eq. (7), so
one does not need to add any extra term to the total energy expression, Eq.
(2) –nor to Eq. (5).

$\Lambda=1.25$ $L=6493$ $L_{\rm cap}=6224$ $N=1500$
---
$n_{v}$ |  $E$ (K) |  $E_{\rm kin}$ (K) |  $\omega$ (K/$\hbar$) |  $a_{x}$ (Å) |  $b_{y}$ (Å) |  $c_{z}$ (Å)
0 | -7658.10 | 519.11 | 0.1001 | 48.41 | 13.63 | 15.40
1 | -7641.16 | 583.36 | 0.1025 | 46.68 | 15.80 | 14.52
2 | -7626.09 | 648.30 | 0.1153 | 44.44 | 17.15 | 16.33
3(L) | -7601.71 | 714.49 | 0.1197 | 42.51 | 18.72 | 15.93
4(L) | -7567.76 | 763.02 | 0.1222 | 42.13 | 19.28 | 16.86
Table 4:  Energetics and morphology of prolate 4He1500 droplets at
$\Lambda=1.25$ for $n_{v}=0-4$. $L$ indicates a linear vortex array. The entry
in boldface corresponds to the globally stable configuration.  $\Lambda=1.25$
$L=133307$ $L_{\rm cap}=127937$ $N=20{,}000$
---
$n_{v}$ |  $E$ (K) |  $E_{\rm kin}$ (K) |  $\omega$ (K/$\hbar$) |  $a_{x}$ (Å) |  $b_{y}$ (Å) |  $c_{z}$ (Å)
0 | -125806.09 | 2883.61 | 0.02781 | 114.26 | 32.39 | 36.54
1 | -125789.56 | 3097.10 | 0.02953 | 111.73 | 35.85 | 36.16
2 | -125789.99 | 3293.39 | 0.03089 | 108.35 | 37.43 | 38.51
3 | -125767.52 | 3494.95 | 0.03203 | 105.28 | 39.80 | 38.28
4∗(C) | -125701.35 | 3676.09 | 0.03306 | 103.20 | 42.47 | 38.92
4(L) | -125714.08 | 3705.57 | 0.03283 | 102.53 | 41.98 | 39.20
5 | -125641.07 | 3887.18 | 0.03371 | 100.35 | 44.39 | 39.88
Table 5: Energetics and morphology of prolate 4He droplets with $N=20{,}000$
at $\Lambda=1.25$ for $n_{v}=0-5$. $L$ indicates a linear, and $C$ a cross-
shaped vortex array. The entry in boldface corresponds to the globally stable
configuration, and the asterisk identifies the highest energy $n_{v}=4$
configuration.

A difficulty inherent to the study of helium droplets is that configurations
with rather distinct morphologies may have similar energies; therefore, a
careful analysis is required in order to distinguish the global minimum from
metastable configurations. An added challenge is that most vortex array
configurations are very robust and, in spite of not being a physically
conserved quantity, the starting $n_{v}$ value is often conserved instead of
being relaxed to the optimal value corresponding to the global equilibrium
configuration. This forces to explore, for a given $N$-atoms droplet, all
possible $n_{v}$ values compatible with the chosen $L$ —a task which becomes
increasingly cumbersome as $L$ increases. This robustness has however the
benefit that one has access to a series of excited configurations, which might
be experimentally accessible, characterized not only by the strictly conserved
$N$ and $L$ values but also by the number of vortices $n_{v}$. We have always
checked the stability of the vortex array against shape distortions.

$\Lambda=1.25$ $L=256093$ $L_{\rm cap}=245756$ $N=35{,}000$
---
$n_{v}$ |  $E$ (K) |  $E_{\rm kin}$ (K) |  $\omega$ (K/$\hbar$) |  $a_{x}$ (Å) |  $b_{y}$ (Å) |  $c_{z}$ (Å)
0 | -225315.35 | 4174.71 | 0.02102 | 137.65 | 38.98 | 43.97
1 | -225307.23 | 4450.00 | 0.02225 | 134.88 | 42.80 | 43.50
2 | -225323.77 | 4695.91 | 0.02318 | 131.14 | 44.40 | 46.30
3 | -225312.04 | 4950.83 | 0.02402 | 127.81 | 47.07 | 46.14
4∗(C) | -225239.14 | 5198.51 | 0.02477 | 125.33 | 50.06 | 46.70
4(L) | -225263.35 | 5217.47 | 0.02464 | 124.69 | 49.46 | 47.12
5(P) | -225185.14 | 5466.69 | 0.02529 | 122.15 | 52.15 | 47.86
5∗(L) | -225173.77 | 5489.12 | 0.02505 | 122.16 | 51.69 | 45.90
6 | -225104.71 | 5713.58 | 0.02600 | 118.84 | 54.80 | 48.43
Table 6: Energetics and morphology of prolate 4He droplets with $N=35{,}000$
at $\Lambda=1.25$ for $n_{v}=0-6$. $L$ indicates a linear, $C$ a cross-shaped,
and $P$ a pentagonal vortex array. The 6-vortex array configuration is a
stretched hexagon. The entry in boldface corresponds to the globally stable
configuration, and the asterisk identifies the highest energy configuration
among those with the same $n_{v}$ value.

For every stationary configuration obtained by solving Eq. (5), a “sharp
density surface” is determined by calculating the locus at which the helium
density equals $\rho_{0}/2$, where $\rho_{0}$ is the atom density of the
liquid; for a spherical distribution this corresponds to the sphere of radius
$R$ defined below. In the case of deformed droplets, three lengths (half-axes)
$a_{x}$, $b_{y}$ and $c_{z}$ are introduced representing the distances from
the center of mass of the droplet to the sharp surface along the principal
axes of inertia. These lengths are represented in the inset in Fig. 1. For an
axisymmetric droplet, $a_{x}=b_{y}\neq c_{z}$. These lengths have been used to
characterize the droplet shape by defining two dimensionless aspect ratios,
$a_{x}/c_{z}$ and $b_{y}^{3}/V$,Bal15 ; Anc18 ; Lan18  where $V$ is the volume
of the non-rotating spherical droplet.

Figure 1:  Aspect-ratio $b_{y}^{3}/V$ vs. $a_{x}/c_{z}$ curve. Black
triangles: 3-vortex 4He1500 configurations.Anc18  Green diamonds: vortex-free
(capillary wave) 4He1500 configurations.Anc18  Magenta squares (circles):
$N=20{,}000\,(35{,}000)$ configurations. The numbers close to the symbols
indicate the $n_{v}$ value. The vertical dotted line separates the
$N=20{,}000$ and 35,000 droplet configurations with $\Lambda=0.8$ (left) from
those with $\Lambda=1.25$ (right). Solid symbols represent the global
equilibrium configurations displayed in the tables. Black starred symbols are
the experimental results of Ref. Lan18, , and blue dots connected by a dashed
line are the classical rotating drop results.But11 ; Bal15  The inset displays
the lengths $a_{x},b_{y}$ and $c_{z}$ used to characterize the droplet shape.

Figure 2:  Dimensionless angular velocity $\Omega$ vs. dimensionless angular
momentum $\Lambda$. Black triangles: 3-vortex 4He1500 configurations.Anc18
Green diamonds: vortex-free (capillary wave) 4He1500 configurations.Anc18  The
oblate and prolate configurations connected by a red horizontal line at
$\Omega=0.5247$ are the twin configurations we refer to in the text. Magenta
squares (circles) represent the $N=20{,}000\,(35{,}000)$ results for
$n_{v}=0,2$, and 5 configurations (for given $N$ and $\Lambda$ values, the
larger the $n_{v}$, the bigger the $\Omega$). For the sake of completeness, we
also include the critical points for nucleating a single vortex (red symbols):
triangle, $N=1500$; asterisk, $N=5000$;Cop19  square, $N=20{,}000$; circle,
$N=35{,}000$. The red diamond at $(\Lambda,\Omega)=(1.4\times
10^{-2},4.6\times 10^{-2})$ is critical point for nucleating a vortex for
$N=10^{9}$ in the hollow core model. The blue dots connected by a dashed line
are the classical rotating drop results.But11 ; Bal15

Figure 3:  Densities on the symmetry plane perpendicular to the rotational
axis for the $N=35{,}000$ He droplet hosting zero, one, and and $n_{v}=3-6$
vortices at $\Lambda=0.80$. The streamlines of the superfluid flow are
superimposed and their direction is counterclockwise. The positions of the
vortex cores are marked by red dots for visualization. The color bar shows the
atom density in units of Å-3, and the black bar represents a distance of 100
Å.

The shapes of classical drops exclusively subject to surface tension and
centrifugal forces follow a universal stability diagram in terms of the
dimensionless angular momentum $\Lambda$ and angular velocity $\Omega$ defined
byBro80 ; But11

$\begin{split}\Omega&\equiv\sqrt{\frac{m\,\rho_{0}\,R^{3}}{8\,\gamma}}\;\omega\\\
\Lambda&\equiv\frac{\hbar}{\sqrt{8\gamma R^{7}m\rho_{0}}}\,L\end{split}$ (8)

where $m$ is the atom mass and $\gamma$ is the surface tension of the liquid.
For liquid 4He at zero temperature and pressure, $\gamma$ = 0.274 K Å-2,
$\rho_{0}$ = 0.0218 Å-3, the sharp radius of the spherical droplet is given by
$R=r_{0}N^{1/3}$ with $r_{0}=2.22$ Å,Bar06 , and $\hbar^{2}/m$ = 12.12 K Å2.
We recall that $\hbar=7.6382$ K ps and that $\hbar c=2.2899\times 10^{7}$ K Å.

We want to stress that this universality is lost if one goes beyond that
simple model, e.g., by using a DFT description of the liquid, since this
incorporates effects such as surface diffuseness and liquid
incompressibility:Pi20  although such effects are expected to be less
important as the droplet size increases, they can be tangible for the droplet
sizes investigated here. Yet, Eqs. (8) are very useful as they allow for a
comparison between experimental and calculated droplets, even if the latter
are much smaller (for obvious computational reasons) than those probed in the
experiments. However, one should have in mind that some differences may appear
in the comparison due to these unavoidable finite-size effects. Besides, the
presence of vortices in the case of superfluid droplets definitely breaks the
universality of the stability diagram, as shown below.

##  III Results and discussion

Systematic DFT calculations are very cumbersome to carry out, especially when
several configurations hosting different numbers of vortices have to be
analyzed and their geometries have to be tested against distortions of the
vortex array. This hinders a systematic exploration of the phase diagram. For
this reason, we have limited our study to some relevant cases which embody the
physics we aim to discuss. We have focused in particular on two $\Lambda$
values that embrace a fairly large range of prolate configurations,Anc18
namely $\Lambda=0.80$ and 1.25, for droplets with $N=1500$ atoms (a value for
which a rather detailed series of DFT calculations is availableAnc18 ) and
with $N=20{,}000$ and 35,000; these large prolate droplets (for DFT standards)
are well suited to disclose the interplay between capillary waves and vortex
arrays object of this study, and how the number of vortices in the globally
stable configuration evolves as a function of $N$.

Tables 1–6 collect the results obtained for the mentioned $(\Lambda,N)$ values
(one pair of values per table) and give details on their energetics and
morphology. In each table we have written in boldface the entry corresponding
to the globally stable configuration, i.e., that with the lowest energy for
the given values of $N$ and $\Lambda$. When more than one configuration has
been found for a given number of vortices $n_{v}$, the more energetic one is
identified with an asterisk (∗). Triangular, cross-shaped, pentagonal and
linear vortex arrays are denoted by $T,C,P,$ and $L$, respectively. Notice
that the configurations in a given table can be directly compared, as they
correspond to the same values of two strictly conserved quantities in isolated
droplets.

As already discussed in the Introduction Section, it appears from the Tables
that for vortex-free superfluid droplets the intermediate half-axis $b_{y}$ is
aligned with the rotational axis, at variance with the classical rotating
droplets where the shortest half-axis $c_{z}$ is aligned instead with the
rotational axis. We are not aware of a general demonstration of this counter-
intuitive result; its proof is however known for ellipsoidal drops made of
irrotational (potential) fluids.Lam45  This peculiarity of spinning
irrotational drops has to be taken into account for a proper analysis of the
experiments: identifying the shortest axis of a spinning superfluid 4He
droplet with the rotational axis might be incorrect. For vortex-hosting
superfluid droplets, though, the rotational axis coincides most of the times
with the shortest ($c_{z}$) one, as for viscous drops. But there are
exceptions, e.g., for $N=20{,}000$ and 35,000 when $n_{v}=2$.

In most cases, the presence of a vortex array confers to the droplet the
appearance of a rotating viscous droplet. A notorious example of this
apparently classical behavior is the meniscus that develops, at the liquid-
vapor interface, in a rotating bucket filled with superfluid helium above the
critical angular velocity required for vortex nucleation.Osb50 ; Don91  Hence,
determining the angular momentum of the droplet (magnitude and direction) and
whether it hosts a vortex array or not seems unavoidable before drawing a
definite conclusion about how superfluid helium droplets rotate.

Figure 4:  Densities on the symmetry plane perpendicular to the rotational
axis for the $N=35{,}000$ He droplet hosting zero, one, and $n_{v}=3-6$
vortices at $\Lambda=1.25$. The streamlines of the superfluid flow are
superimposed and their direction is counterclockwise. The positions of the
vortex cores are marked by red dots for visualization. The color bar shows the
atom density in units of Å-3, and the black bar represents a distance of 100
Å.

We show in Fig. 1 the dimensionless ratios $b_{y}^{3}/V$ versus $a_{x}/c_{z}$
for the configurations collected in the tables, irrespective of whether they
correspond to globally stable or metastable configurations. The $N=1500$
results have been complemented with those shown in Fig. 1 of Ref. Anc18, ,
which presents data for a large sample of $\Lambda$ values. Also shown are the
experimental results,Lan18  which have an average atom number $\langle
N\rangle=6\times 10^{9}$, and the result for classical rotating drops.Bal15
While the results for vortex-hosting configurations are similarly distributed
and roughly follow the classical drops trend, the figure shows that vortex-
free prolate configurations form a separate branch (capillary branch)
displaying a completely different behavior. Both branches meet at the
$(a_{x}/c_{z},b_{y}^{3}/V)=(1,3/4\pi)$ point corresponding to spherical
droplets.

It is worth noticing that configurations with the same $n_{v}$ but rather
different $N$ values yield similar points on the $(a_{x}/c_{z},b_{y}^{3}/V)$
plane, especially when the vortex array has the same morphology (linear,
cross, $\ldots$), and in particular for vortex-free configurations. The
universality of the classical results, which allows to scale them with $N$, is
not completely lost. Unfortunately, even $N=35{,}000$ is too small a value
compared to the experimental ones to allow us to draw any sensible conclusion
about this issue. The figure also shows that, for a fixed $\Lambda$ value, the
larger the $n_{v}$ the more similar viscous and vortex-hosting droplets are.

We thus see that knowledge of the shape of a large number of deformed
configurations is not enough to unambiguously characterize their rotational
behavior as classical or superfluid, and that one has to combine this
information with the simultaneous determination of the angular momentum of the
droplet by, e.g., obtaining the stability diagram in the $(\Lambda,\Omega)$
plane. This diagram is shown in Fig. 2 for some of the configurations in Fig.
1. While the capillary branch may arrive up to nearly $\Lambda=0$, as only a
small deformation is needed for the droplet to sustain capillary waves, the
vortex-hosting branch must abruptly interrupt at some
$(\Lambda_{c},\Omega_{c})$ point, as a critical angular velocity $\omega_{c}$
is needed for nucleating a vortex line in a superfluid droplet.Pit16  The red
symbols in Fig. 2 represent the critical vortex nucleation point for the three
$N$ values used in this study. For the sake of completeness, we also display
the DFT result for $N=5000$,Cop19  and that of the hollow core modelDal00  for
a $N=10^{9}$ drop taking for the vortex core radius a value of 1Å.

The existence of this critical point dramatically distinguishes superfluid
from viscous droplet rotational behavior, as no axisymmetric superfluid He
droplet can be set into rotation for $\Lambda<\Lambda_{c}$. Another
conspicuous difference between viscous and superfluid vortex-hosting droplet
results can be observed in Fig. 2, namely the location of the oblate-to-
prolate bifurcation point. For viscous droplets, it is at $\Lambda=1.2$,But11
; Hei06  whereas for droplets of the size studied here, DFT yields the
bifurcation slightly below $\Lambda=0.8$. Indeed, the three-vortex 4He1500
configurations connected by a red horizontal line correspond to oblate (left)
and a prolate (right) configurations. We attribute this large difference to a
finite size effect, although we cannot ascertain this. In Fig. 2, finite size
effects also clearly affect the position of the critical vortex nucleation
point.

Comparing the results for the same $N$ value, one can see from the tables that
the ratio $b_{y}/a_{x}$ decreases as $\Lambda$ increases. This causes, e.g.,
that for the 4He1500 droplet, the most stable $n_{v}=4$ vortex array, that was
cross-shaped at $\Lambda=0.80$, is instead linear at $\Lambda=1.25$; the same
happens for $N=20{,}000$. Large droplets can host both kinds of vortex
arrangements, see, e.g., Table 6. For a given $\Lambda$, a large droplet may
accommodate a non-linear vortex array more easily. Notice for instance that,
at $\Lambda=1.25$, the cross-shaped $n_{v}=4$ configuration for $N=35{,}000$
is still more stable than the linear one. The existence of multiple
configurations for the same $n_{v}$ value is also ubiquitous in axisymmetric
configurations,Cam79 ; Anc15 ; Anc14  making the search for the lowest energy
configuration very time-consuming.

Further inspection of the tables shows that, for given $N$ and $\Lambda$
values, the globally stable spinning prolate configurations are not
necessarily vortex-free, and that, contrarily to a naive expectation, for
large, fixed $N$ and $\Lambda$ values, the energy is not monotonously varying
with $n_{v}$, see for instance Tables 1, 2 and 6. We have found that for the
largest $\Lambda$ value considered in this study ($\Lambda=1.25$), the vortex-
hosting configuration becomes the globally stable configuration only above a
“critical” $N$ value. We have checked that for $N=30{,}000$ the vortex-free
and the $n_{v}=2$ configurations are nearly degenerate with energies of
$-191\,994.86$ K and $-191\,994.49$ K, respectively. As Table 6 shows, the
globally stable configuration of the $N=35{,}000$ droplet is the $n_{v}=2$
one. It is thus very likely that the prolate large $N$ drops studied in the
FELGom14  and XUVLan18  experiments contain both vortex arrays and capillary
waves.Oco20  Notice also that the energy differences between the first excited
configurations and the globally stable ones are small, and it cannot be
completely discarded that some of such excited configurations are also
detected in the experiments. These considerations indicate that vortex-free
prolate configurations might be extremely difficult to detect either because
they correspond to globally stable configurations only for small-$N$ droplets
that escape detection in current diffractive imaging experiments, or because
they are metastable configurations in the case of large-$N$ drops and likely
decay to vortex-hosting configurations before they are probed.

Figures 3 and 4 show some 2D densities corresponding to the $N=35{,}000$
droplet. One may see that, in order to store a large angular momentum, a
vortex-free configuration has to be very stretched –see the discussion on
capillary waves below. This is a general trend that implies an increase of the
droplet surface energy; as shown in the tables, the kinetic energy also
increases with the number of vortices. The globally stable configuration is
determined by a delicate balance between these two competing effects. Figure 4
shows that the $N=35{,}000$ droplet is large enough to accommodate up to 6
vortices, and that the shape of the droplet evolves from two-lobed to a more
compact, “pill-shaped” appearance as found in the experiments.Ber17 ; Lan18 ;
Oco20  Despite most of these configurations are metastable, these shape
transitions exemplify the competition between compact and more linear vortex
arrays arrangements to determine the appearance of the droplet. Notice that
Fig. 1 also displays the same effect: as $n_{v}$ increases for a given
$\Lambda$, the point representing the vortex-hosting configuration
dramatically displaces to smaller $a_{x}/c_{z}$ values, yielding at the same
time a larger $b^{3}_{y}/V$, hence more compact droplet shapes. It is also
worth seeing from the 2D figures that two-lobed configurations as those
reported in Refs. Ber17,  and Lan18,  appear when $n_{v}$ is small; increasing
$n_{v}$ reduces the wrist between the lobes, as the vortex array becomes less
linear and more similar to a patch of a triangular arrangement (as expected
when $n_{v}$ becomes very large). Since the number of pill-shaped droplets
found in the experimentLan18  is six times larger than that of dumbbell-like
droplets, this would again indicate that large, very prolate droplets host
fairly large vortex arrays.

Figure 5 displays a side view of the $N=35{,}000$ droplet at $\Lambda=1.25$
for the $n_{v}=0-3$ configurations. One can see how vortices locally distort
the droplet surface as they have to hit it perpendicularly. A similar figure
can be found in Ref. Anc15,  for oblate configurations. Thus, droplets hosting
large vortex arrays are far from being ellipsoidal. In the diffractive imaging
experiments, the images used to identify the presence of vortices with a well
oriented direction of their coresOco20  would correspond to the flat surfaces
limiting from above and below the droplet side view shown in this figure.

Vortex arrays in prolate configurations are distorted, as found in the
experimentsOco20  and displayed in the 2D figures. To characterize these
distortions, we have calculated an oblate configuration “twin” of the prolate
$N=1500$, $\Lambda=0.80$, $n_{v}=3$ configuration for the same value of
$\Omega$, namely 0.5247, which implies a lower $\Lambda$ value, see Fig. 2.
For this $n_{v}=3$ oblate configuration we have found $\Lambda=0.714$,
$a_{x}/c_{z}=1.38$, and $b_{y}^{3}/V=0.318$, the latter two values to be
compared to those of its twin prolate configuration, $a_{x}/c_{z}=1.68$ and
$b_{y}^{3}/V=0.207$. In the oblate $n_{v}=3$ configuration the vortex array
forms an equilateral triangle with an inter-vortex distance of $16.6$ Å. In
the $n_{v}=3$ prolate configuration the triangle is isosceles and stretched in
the direction of the largest principal axis of inertia ($x$ axis). The
stretched inter-vortex distance is 22.1 Å, and the other two sides of the
triangle correspond to an inter-vortex distance of 15.4 Å.

Figure 5:  Densities on the symmetry plane containing the vortex lines for the
$N=35{,}000$ droplet hosting $n_{v}=0-3$ vortices at $\Lambda=1.25$. The color
bar shows the atom density in units of Å-3, and the black bar represents a
distance of 100 Å.

All configurations displayed in the 2D figures are stationary in the
corotating frame; as a consequence, they would be seen from the laboratory
frame as if they were rotating, apparently like a rigid-body, with the angular
frequency $\omega$ imposed to obtain them. Let us recall and stress that this
is a misleading appearance. The motion of a fluid is a combination of
translation, rotation and deformation of the fluid elements, and only when the
vorticity (defined as he curl of the velocity field of the fluid) is non zero,
one may speak of a true rotation, being important to distinguish between
rotation and motion of the fluid elements along a curved path,Tri82 ; Don91
e.g., around a vortex core. Since the velocity field of the superfluid is
potential, vorticity is zero except on the vortex lines; in their absence,
vorticity is zero everywhere.

Figure 6: Trajectories of the vortex cores $(x_{c},y_{c})$ on the $z=0$ plane
corresponding to the prolate $N=1500$, $\Lambda=0.80$, $n_{v}=3$
configuration. One vortex is turning around the center of mass of the droplet
(shown by a cross) at a distance of 7.7 Å, while the other two vortices rotate
at a distance of 11.0 Å. The inset represents the droplet density on the
symmetry plane perpendicular to the rotational axis.

To illustrate this situation, we have simulated within the TDDFT approachAnc17
the time evolution of the prolate 4He1500 droplet for the vortex-free and the
$n_{v}=3$ configurations at $\Lambda=0.8$. Fig. 6 shows, for the $n_{v}=3$
case, the trajectory $(x_{c},y_{c})$ of the vortex cores on the $z=0$ plane.
It can be observed that the inter-vortex distances do not change during the
evolution, indicating that the three-vortex array rotates with the angular
velocity imposed to the corotating frame, $\omega=1.871\times 10^{-2}$ ps-1.
We see this dynamics as the rigid rotation of the vortex cores (which is not
in contradiction with the previous discussion, as they are empty), around
which the irrotational superflow accommodates in its displacement. One may
also notice the accuracy of the stationary solution we have obtained;
otherwise, the vortex core trajectories, plotted here point by point, would
not be the perfect circumferences displayed in Fig. 6. The figure also
exemplifies the robustness of the fixed-$n_{v}$ configurations mentioned in
Sec. II. These configurations –which are stationary in the corotating frame–
likely correspond to local energy minima and are separated by an energy
barrier from other configurations having a different $n_{v}$ but the same $N$
and $\Lambda$ values.

The interested reader may find two videos in the Supplemental MaterialSM
showing the time evolution of these two configurations in the corotating frame
for about 450 ps. It may be illuminating to look at the time evolution of the
streamlines pattern: it shows how different the apparent rotation of the
superfluid droplet is, compared to the true rotation of a viscous drop. We
want to stress that the energy, particle number and angular momentum of the
isolated droplet are strictly conserved during the evolution.

It is worth recalling that vorticity in quantum fluids may often lead to a
turbulent regimeTsu09 ; Nem13  like, e.g., that resulting from the decay of
vortex tangles in bulk liquid 4He, or the turbulence in Bose Einstein
condensates (BEC) generated by fast stirring the condensate using laser beams.
Our vortex structures, however, are stable against decay, reconnections or
other mechanisms which are known to lead to turbulent behavior in liquid 4He
or BECs. This is what we indeed observe in our simulations: in spite of the
possibility of bending and displacing of the vortex cores, which is allowed by
the three-dimensional nature of our calculations, the vortex structures we
find are either stationary states in the co-rotating system or, when let to
evolve under real-time dynamics, they rotate around the central axis but
remain otherwise stable, without showing any tendency to be expelled or
reconnect with nearby vortices. This is in agreement with the experimental
measurements on nanoscopic 4He droplets hosting vortices where no sign of
decay of the vortex structures is observed.Oco20  Turbulent behavior may arise
though in nanoscopic 4He droplets under special conditions, as illustrated in
Ref. Esc19, , where a TDDFT study of the merging of two vortex-free superfluid
4He droplets has shown that vortex rings are dynamically nucleated during the
merging process and that their annihilation produces a massive emission of
rotons and subsequent turbulent behavior, whose scaling follows the very
general power laws underlying turbulence in quantum fluid systems.Tsu09 ;
Nem13

Streamlines of the superfluid flow are shown in Figs. 3 and 4. The superflow
follows the direction of the angular frequency imposed to the corotating frame
(counterclockwise in this case). Streamlines allow to infer by visual
inspection the coexistence of vortices and surface capillary waves, as their
velocity fields are very different. The streamlines associated to vortices
wrap around their cores, whereas those associated to capillary waves end
abruptly at the droplet surface.Fet74 ; Cop17 ; Oco20  As already mentioned,
separating the contribution of capillary waves and vortices in the velocity
fields is generally not possible, the superfluid velocity field being
proportional to the gradient of the phase $\cal{S}(\mathbf{r})$ of the
effective wave function $\Psi({\mathbf{r}})$, where the contributions of both
vortices and capillary waves are entangled.

The coexistence of one single vortex and a quadrupolar irrotational flow in an
elliptic cylinder filled with helium rotating at constant $\omega$ was studied
by Fetter,Fet74  who anticipated the possible appearance of a row of vortices
along the longer axis of the elliptic vessel cross section. At variance with
isolated droplets, the cylinder has a rigid surface of known geometry. Writing
the phase of $\Psi({\mathbf{r}})$ as the sum of a term arising from the vortex
line and another from the quadrupolar flow (the same guess that we made in Eq.
(6), but in our case only for the starting configuration), he could determine
how the angular momentum is shared between the vortex and the quadrupolar
irrotational flow. It does not look feasible to extend the cylinder approach
to the $n_{v}>1$ case. Even for one single vortex, the study of isolated
droplets is much more complex, as their shapes are not ellipsoidal in general
and they are only known after the stationary configuration has been determined
by solving Eq. (5).

Yet, to analyze the experimental resultsOco20  it has been useful to know, at
least approximately, how angular momentum is shared between vortices and
capillary waves. In order to do that, we computed the deformation parameter
$\varepsilon$ defined in Eq. (1) using the $a_{x}$ and $b_{y}$ parameters
given in the tables for the $n_{v}=0$ configurations, and the relation giving
the angular momentum associated to capillary waves, $L_{\rm
cap}=\varepsilon^{2}\,\Theta_{\rm rig}\,\omega$ (see the discussion following
Eq. (1)).

We have found that it works surprisingly well, as it coincides with the exact
$L$ calculated within the DFT approach to within a few percent. The $L_{\rm
cap}$ value obtained for each studied $n_{v}=0$ configuration is given in the
tables. For vortex-hosting configurations, the angular momentum stored in the
vortex arrays may then be estimated subtracting $L_{\rm cap}$ from the total
$L$. The validity of this procedure cannot be assessed though. So far, the
only possibility to experimentally determine the angular momentum of the drop
has been to use Feynman’s formulaFey55  to obtain the angular velocity
$\omega$ from the vortex density, which could be determined from the 2D
droplet images, and the above expression for $L_{\rm cap}$, see Ref. Oco20,
for details.

##  IV Summary

We have shown that the presence of vortex arrays in spinning superfluid 4He
droplets and their interplay with capillary waves has a profound influence in
the determination of their globally stable configuration. The presence of
vortex arrays, irrespective of whether they are detected or not,Anc18  is the
only plausible explanation for the existence of oblate configurations, which
otherwise will be in conflict with quantum mechanics or would imply that
droplets are not superfluid, which is extremely unlikely due to the working
temperatures in 4He droplets experiments.Toe04

Surface capillary waves are ubiquitous in prolate superfluid 4He droplets.
Depending on their size and angular momentum, these waves may coexist with
vortex arrays. We have found that the global equilibrium configuration of
small prolate droplets is vortex-free, but the situation changes as the
droplet size increases. This result is in agreement with a recent
experiment,Oco20  where it has been disclosed that vortex arrays and capillary
waves coexist in very large drops.

While vortex arrays may be detected by doping the droplets, capillary waves
are very elusive and escape direct detection. This poses a serious difficulty
to determine the angular momentum of the superfluid droplet. From the
theoretical side, microscopic approaches such as DFT treat vortex arrays and
capillary waves on the same footing, and one has to resort to approximate
methods to disentangle their contribution to the total angular momentum of the
droplet. Determining –even in a model dependent way– the angular momentum of
the droplets is essential if one wants to characterize spinning superfluid
droplets, as only simultaneous knowledge of the droplet morphology and angular
momentum allows for a sensible comparison with classical or quantum models.

Contrarily to viscous droplets executing rigid body rotation (modeled by
viscous droplets subject to surface tension and centrifugal forces alone), the
stability phase diagram of superfluid droplets is not universal and cannot be
characterized by dimensionless angular momentum $\Lambda$ and dimensionless
angular velocity $\Omega$ as its classical counterpart. Their knowledge does
not determine univocally the rotational properties of superfluid helium
droplets, which display a clear dependence on the droplet size and/or the
number of vortices they host. This is not in contradiction with the recent
findingOco20  that big superfluid 4He droplets hosting a large number of
vortices seem to rotate like rigid-bodies, thus following the classical
stability phase diagram. Rather, it is a manifestation of finite size effects
which still have to be studied in detail and, together with recent works on
rotating 3He dropletsPi20 ; Ver20  and mixed 3He-4He droplets,But20 ; Pib20
call for further experimental research.

###### Acknowledgements.

We are indebted to Andrey Vilesov for useful discussions. We thank Thomas
Möller and Bruno Langbehn for providing us with the results of Ref. Lan18,
shown in Fig. 1, and Sam Butler for providing us with the results of the
classical model calculations used in this work. This work has been performed
under Grant No FIS2017-87801-P (AEI/FEDER, UE). J.M.E. acknowledges support
from the Spanish Research Agency (AEI) through the Severo Ochoa Centres of
Excellence programme (grant SEV-2017-0706) and from the European Union MaX
Center of Excellence (EU-H2020 Grant No. 824143).

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