The energy density functional of a uniform mixture in the mean field approximation is a quadratic form [@stringari]: $$\label{MF}
\epsilon_\mathrm{MF} =\frac12 g_{11} n_1^2 + \frac12 g_{22} n_2^2 + g_{12} n_1 n_2,$$ where $n_1$ and $n_2$ are densities of the species. The mixtures can be miscible if $|g_{12}| < \sqrt{g_{11}g_{22}}$, or immiscible if interspecies repulsion dominates, $g_{12}> \sqrt{g_{11}g_{22}}$. On the contrary, if inter-species attraction is strongly attractive, $g_{12} < - \sqrt{g_{11}g_{22}}$, a mixture collapses. Typically, miscible mixtures have to be kept in external traps since, if left alone, they expand to minimize their energy.

The mean-field description overlooks existence of ultra-dilute quantum droplets — the exotic phases of the self-bound incompressible system of a two component Bose-Einstein condensates (BECs), stabilized by quantum fluctuations [@Petrov15], and with densities orders of magnitude smaller than of ordinary liquids.

In a weakly interacting regime, the energy related to the quantum fluctuations is small, and, for a single-component BEC, is known as the Lee-Huang-Yang (LHY) correction to the ground state energy of the system [@Lee57]: $$\begin{aligned}
\epsilon_\mathrm{LHY} =   \frac{128}{30\sqrt{\pi}}\, gn^2 \sqrt{na^3},\end{aligned}$$ where $n$ is the density, the coupling strenght $g=4 \pi \hbar^2 a/m$, $a$ is the positive $s$-wave scattering length, and $m$ is the atomic mass. 