Rapidly-rotating quantum droplets confined in a harmonic potential

We consider a "symmetric" quantum droplet in two spatial dimensions, which rotates in a harmonic potential, focusing mostly on the limit of "rapid" rotation. We examine this problem using a purely numerical approach, as well as a semi-analytic Wigner-Seitz approximation (first developed by Baym, Pethick et al.) for the description of the state with a vortex lattice. Within this approximation we assume that each vortex occupies a cylindrical cell, with the vortex-core size treated as a variational parameter. Working with a fixed angular momentum, as the angular momentum increases and depending on the atom number, the droplet accommodates none, few, or many vortices, before it turns to center-of-mass excitation. For the case of a "large" droplet, working with a fixed rotational frequency of the trap $\Omega$, as $\Omega$ approaches the trap frequency $\omega$, a vortex lattice forms, the number of vortices increases, the mean spacing between them decreases, while the "size" of each vortex increases as compared to the size of each cell. In contrast to the well-known problem of contact interactions, where we have melting of the vortex lattice and highly-correlated many-body states, here no melting of the vortex lattice is present, even when $\Omega = \omega$. This difference is due to the fact that the droplet is self-bound. Actually, for $\Omega = \omega$, the "smoothed" density distribution becomes a flat top, very much like the static droplet. When $\Omega$ exceeds $\omega$, the droplet maintains its shape and escapes to infinity, via center-of-mass motion.