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

Within density functional theory, we have studied the interplay between vortex arrays and capillary waves in spinning prolate $^{4}\mathrm{He}$ droplets made of several thousand helium atoms. Surface capillary waves are ubiquitous in prolate superfluid $^{4}\mathrm{He}$ 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 $^{4}\mathrm{He}$ 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.