The density-functional theory of quantum droplets

In quantum droplets, the mean-field energy is comparable to the Lee-Huang-Yang (LHY) energy. In the Bogoliubov theory, the LHY energy of the quantum droplet has an imaginary part, but it is neglected for practical purposes. So far, most theoretical studies of quantum droplets have been based on the extended Gross-Pitaevskii (GP) equation obtained by adding the LHY energy to the GP equation. In this article, we present the density-functional theory of quantum droplets. In our approach, the quantum fluctuations in quantum droplets, as described by an effective action, generate the correlation energy which is real and can be determined self-consistently. Our results are consistent with the extended GP equation in the dilute limit, and the implications to current experiments on quantum droplets are discussed.