Quantum droplets with magnetic vortices in spinor dipolar Bose-Einstein condensates

Motivated by the recent experimental realization of a Bose-Einstein condensate (BEC) of europium atoms, we investigate the self-bound droplet state of a europium BEC with spin degrees of freedom. Under a sufficiently weak magnetic field, the droplet has a torus shape with circulating spin vectors, which is referred to as a magnetic vortex. The ground state transforms from the torus to cigar shape through bistability with an increase in the magnetic field. Dynamical change of the magnetic field causes the torus to rotate due to the Einstein-de Haas effect. The magnetic vortices form a supersolid in a confined system.