in perpendicular direction(s) by $L_{\perp}$.

At this stage, we do not assume any particular geometry yet. The LHY energy density reads: $$\label{LHY_mix}
 \frac{\varepsilon_0}{L^3}  e_\mathrm{LHY} = \lim_{r \rightarrow 0} \frac{\partial }{\partial r} \left( r \frac{1}{2V}\sum_{{\bf k}}e^{i{{\bf k}}{{\bf r}}} 
( \varepsilon_{{\bf k}}- A_{{\bf k}}) \right),$$ where $\varepsilon_{{\bf k}}= \sqrt{E_{k}^2 + 2 E_k(g_{11}n_1+g_{22}n_2) }$ and $ A_{{\bf k}}= E_k + g_{11}n_1+g_{22}n_2$, and $E_k= (\hbar^2k^2)/2m$. We extracted the prefactor $\varepsilon_0/L^3$ to make $e_\mathrm{LHY}$ dimensionless. This form of the LHY energy results from a regularized pseudopotential [@Lee57], and it is equivalent to the formula used in Ref. [@Petrov15], where the origin of the LHY term is attributed to the zero-point energy of the Bogoliubov vacuum.

In writing Eq. (\[LHY\_mix\]), we made two approximations. First, we set $g_{12}^2 = g_{11}g_{22}$ which is consistent with the previous assumptions that the system is about to collapse. This approximation is not a very restrictive one. Second, we limit the analysis to mixtures of two species with equal masses only. Therefore, the system we consider is, for instance, a mixture of atoms in two different internal spin states [@Cabrera17; @Cheiney18; @Fattori17]. The second approximations is quite restrictive, however. We note that the LHY term is equal to the one of 