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

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