Quantum droplets may form out of a gaseous Bose-Einstein condensate, stabilized by quantum fluctuations beyond mean field. We show that multiple singly-quantized vortices may form in these droplets at moderate angular momenta in two dimensions. Droplets carrying these precursors of an Abrikosov lattice remain selfbound for certain timescales after switching off an initial harmonic confinement. Furthermore, we examine how these vortex-carrying droplets can be formed in a more pertubation-resistant setting, by starting from a rotating binary Bose-Einstein condensate and inducing a metastable persistent current via a non-monotonic trapping potential.The formation of self-bound droplets is a well-known macroscopic phenomenon. For an exemplary droplet of water, stability and shape rely on the balance of effective forces between its constituent particles -attractive ones that keep it together, and repulsive ones that prevent it from collapse. Their interplay defines the droplets' surface tension, stabilizing the system in a metastable state. Such droplets do not only occur at a macroscopic level, but are ubiquitous also in the quantum realm, where nuclei [1] and superfluid helium droplets [2][3][4] are prominent examples. While these are rather dense and strongly interacting many-body systems, recent experiments with ultra-cold quantum gases of bosonic atoms uncovered a novel type of quantum liquid: Self-bound droplets may form out of a gaseous Bose-Einstein condensate (BEC) of dysprosium [5][6][7][8][9] or erbium [10], atomic species that are known for their strong dipolar interactions [11][12][13]. Similar droplet states have more recently also been realized with binary Bose gases of potassium in different hyperfine states [14,15], where the inter-and intracomponent interactions are short-ranged. These quantum droplets can be large, containing thousands of atoms. Importantly, they are very dilute -by more than eight orders of magnitude when compared with liquid helium [14]. While the discovery with dysprosium [5-7] at first came as a surprise, the binary self-bound droplet states were theoretically predicted a year before [16] for a scenario similar to the experiments with potassium, and also in lower dimensions [17]. That higher-order corrections beyond mean field may lead to self-bound states was discussed earlier in a different setting in Refs. [18,19]. For the dipolar or binary self-bound bosonic systems of 14,15] the physical mechanism of droplet formation is based on tuning the interactions in gas such that only a weak effective attraction remains. While in pure mean field this would lead to a collapse of the system, weak first-order corrections to the mean field energy, often referred to as the Lee-Huang-Yang (LHY)correction [20], can become comparable in size and may thus stabilize the system.

We introduce a density functional formalism to study the ground-state properties of stronglycorrelated dipolar and ionic ultracold bosonic and fermionic gases, based on the self-consistent combination of the weak and the strong coupling limits. Contrary to conventional density functional approaches, our formalism does not require a previous calculation of the interacting homogeneous gas, and it is thus very suitable to treat systems with tunable long-range interactions. Due to its asymptotic exactness in the regime of strong correlation, the formalism works for systems in which standard mean-field theories fail.Introduction -In contrast with its widespread use and success in areas as diverse as quantum chemistry [1], materials science [2] or semiconductor nanostructures [3], Density Functional Theory (DFT) has received relatively little attention in the very active field of ultracold atomic gases. It is well known that the Hohenberg-Kohn theorems, originally formulated in terms of the electron gas [4,5], hold for both fermionic and bosonic systems, as well as for interactions different than the Coulomb one. However, the lack of adequate density functionals has hindered the role of DFT in the study of ultracold atomic gases in favour of other well-established approaches, such as the widely used Gross-Pitaevskii (GP) method in the case of Bose gases. The latter is a mean-field approach and does not allow treating the effect of correlations, which play a crucial role in many different phenomena occurring in ultracold quantum gases [6]. One then often turns to configuration-interaction (CI), quantum Monte Carlo (QMC) or Green's-function methods (for recent reviews, see, e.g., Refs. 6-8).

A spin-orbit-coupled Bose-Einstein-condensed cloud of atoms confined in an annular trapping potential shows a variety of phases that we investigate in the present study. Starting with the noninteracting problem, the homogeneous phase that is present in an untrapped system is replaced by a sinusoidal density variation in the limit of a very narrow annulus. In the case of an untrapped system there is another phase with a striped-like density distribution, and its counterpart is also found in the limit of a very narrow annulus. As the width of the annulus increases, this picture persists qualitatively. Depending on the relative strength between the inter-and the intra-components, interactions either favor the striped phase, or suppress it, in which case either a homogeneous, or a sinusoidal-like phase appears. Interactions also give rise to novel solutions with a nonzero circulation.

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