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 a quantum droplet has an imaginary part, which has been neglected in most studies for practical purposes. Thus far, most theoretical studies on quantum droplets have been based on the extended Gross-Pitaevskii (GP) equation, which includes the contribution of the LHY energy to the chemical potential. 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 a real correlation energy that can be determined consistently. Using density-functional theory, we calculated the energy, quantum depletion fraction, and excitations of the droplet. Our results for the ground-state energy and quantum depletion fractions were consistent with the Monte Carlo results. We also discuss the implications of our theory.