Coexistence of vortex arrays and surface capillary waves in spinning prolate superfluid $^{4}\mathrm{He}$ nanodroplets

Coexistence of vortex arrays and surface capillary waves in spinning prolate superfluid ${}^{4}{He}$ nanodroplets

Martí Pi, José María Escartín, Francesco Ancilotto, and Manuel Barranco

Phys. Rev. B 104, 094509 – Published 9 September 2021

Abstract

Within density functional theory, we have studied the interplay between vortex arrays and capillary waves in spinning prolate ${}^{4}{He}$ droplets made of several thousand helium atoms. Surface capillary waves are ubiquitous in prolate superfluid ${}^{4}{He}$ 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 [O'Connell et al., Phys. Rev. Lett. 124, 215301 (2020)] that disclosed that vortex arrays and capillary waves coexist in the equilibrium configuration of very large drops. In contrast to viscous droplets executing rigid-body rotation, the stability phase diagram of spinning ${}^{4}{He}$ 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.

Received 15 April 2021
Revised 28 August 2021
Accepted 31 August 2021

DOI:https://doi.org/10.1103/PhysRevB.104.094509

Physics Subject Headings (PhySH)

Quantum fluids & solids Superfluidity Vortices in superfluids

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Martí Pi ^1,2, José María Escartín ³, Francesco Ancilotto ^4,5, and Manuel Barranco^1,2

¹Departament FQA, Facultat de Física, Universitat de Barcelona, Avenida Diagonal 645, 08028 Barcelona, Spain
²Institute of Nanoscience and Nanotechnology, Universitat de Barcelona, 08028 Barcelona, Spain
³Catalan Institute of Nanoscience and Nanotechnology, CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain
⁴Dipartimento di Fisica e Astronomia “Galileo Galilei” and CNISM, Università di Padova, via Marzolo 8, 35122 Padova, Italy
⁵CNR-IOM Democritos, via Bonomea 265, 34136 Trieste, Italy

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Vol. 104, Iss. 9 — 1 September 2021

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Figure 1
Aspect ratio $b_{y}^{3} / V$ vs $a_{x} / c_{z}$ curve. Black triangles: Three-vortex ${}^{4}{He}_{1500}$ configurations [20]. Green diamonds: Vortex-free (capillary wave) ${}^{4}{He}_{1500}$ configurations [20]. 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 $Λ = 0.8$ (left) from those with $Λ = 1.25$ (right). Solid symbols represent the global equilibrium configurations displayed in the tables. Black stars are the experimental results of Ref. [16], and blue dots connected by a dashed line are the classical rotating drop results [31, 32]. The inset displays the lengths $a_{x}, b_{y}$ , and $c_{z}$ used to characterize the droplet shape.
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Figure 2
Dimensionless angular velocity $Ω$ vs dimensionless angular momentum $Λ$ . Black triangles: Three-vortex ${}^{4}{He}_{1500}$ configurations [20]. Green diamonds: Vortex-free (capillary wave) ${}^{4}{He}_{1500}$ configurations [20]. The oblate and prolate configurations connected by a red horizontal line at $Ω = 0.5247$ 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 $Λ$ values, the larger $n_{v}$ is, the bigger $Ω$ is). For the sake of completeness, we also include the critical points for nucleating a single vortex (red symbols): Triangle, $N = 1500$ ; asterisk, $N = 5000$ [26]; square, $N = 20 000$ ; circle, $N = 35 000$ . The red diamond at $(Λ, Ω) = (1.4 \times 10^{- 2}, 4.6 \times 10^{- 2})$ is the 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 [31, 32].
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Figure 3
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 $Λ = 0.80$ . 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 $Å^{- 3}$ ), and the black bar represents a distance of 100 Å.
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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 $Λ = 1.25$ . 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 $Å^{- 3}$ ), and the black bar represents a distance of 100 Å.
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Figure 5
Densities on the symmetry plane containing the vortex lines for the $N = 35 000$ droplet hosting $n_{v} = 0 - 3$ vortices at $Λ = 1.25$ . The color bar shows the atom density (in $Å^{- 3}$ ), and the black bar represents a distance of 100 Å.
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Figure 6
Trajectories of the vortex cores $(x_{c}, y_{c})$ on the $z = 0$ plane corresponding to the prolate $N = 1500$ , $Λ = 0.80$ , $n_{v} = 3$ 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.
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Coexistence of vortex arrays and surface capillary waves in spinning prolate superfluid ${}^{4}{He}$ nanodroplets

Martí Pi, José María Escartín, Francesco Ancilotto, and Manuel Barranco

Phys. Rev. B 104, 094509 – Published 9 September 2021

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Coexistence of vortex arrays and surface capillary waves in spinning prolate superfluid He4 nanodroplets

Martí Pi, José María Escartín, Francesco Ancilotto, and Manuel Barranco

Phys. Rev. B 104, 094509 – Published 9 September 2021

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Coexistence of vortex arrays and surface capillary waves in spinning prolate superfluid ${}^{4}{He}$ nanodroplets