its own external potential an averaged one-body potential induced by the other component. To obtain the effective mean-field Hamiltonian $H_{\sigma}^{MF}$ of species $\sigma$ we need to integrate out the other component $\bar{\sigma}$. For convenience we also subtract the energy offset $c_{\bar{\sigma}}=\braket{N_{\bar{\sigma}} | H_{\bar{\sigma}} | N_{\bar{\sigma}}}$ caused by the one-body energy of component $\bar{\sigma}$: $$\begin{aligned}
\label{eq:MF_hamiltonian}
    \nonumber
    H_{\sigma}^{MF} && = \braket{N_{\bar{\sigma}}|H|N_{\bar{\sigma}}} -
    c_{\bar{\sigma}}
    = H_{\sigma} + N_{\bar{\sigma}} g_{_{MI}} 
    \sum_{i=1}^{N_{\sigma}} \rho_{_{MF}}^{\bar{\sigma}}(z_i) \\
    \nonumber
    && = \sum_{i=1}^{N_{\sigma}} \left(-\frac{1}{2} \frac{\partial^2}{\partial z_i^2}
    + V_{\sigma} (z_i) + V_{\sigma}^{ind} (z_i)\right) \\
    && = \sum_{i=1}^{N_{\sigma}} \left(-\frac{1}{2} \frac{\partial^2}{\partial z_i^2}
    + V_{\sigma}^{eff} (z_i)\right),\end{aligned}$$ where $\rho_{_{MF}}^{\sigma}(z) = |\varphi_{_{MF}}^{\sigma}(z)|^2$ is the one-body density of species $\sigma$ normalized as $\int dz \ \rho_{_{MF}}^{\sigma}(z) = 1$, $V_{\sigma}^{ind} (z)=N_{\bar{\sigma}} g_{_{MI}} \rho_{_{MF}}^{\bar{\sigma}}(z)$ the induced one-body potential, $V_{\sigma}^{eff} (z)=V_{\sigma} (z)+V_{\sigma}^{ind} (z)$ the effective one-body potential and $\bar{\sigma}\neq\sigma$.

To systematically distinguish between different phases we define the following two functions, applicable also in the more general case of a many-body treatment in Sec. \[sec:few\_body\_MLX\]: $$\label{quali1}
    \Delta_{\sigma} = 
    \frac{\rho^{\sigma}_{1} (z=0)}{\underset{z}{\rm{max}} \ \rho^{\sigma}_1 (z)}$$ $$\label{quali2}
    d = |\int_{-\infty}^{\infty} dz \ z \rho_{1}^{M}(z) - 
    \int_{-\infty}^{\infty} dz \ z \rho_{1}^{I}(z)|,$$ with the one-body density $\rho_1^{\sigma}(z)$ of component $\sigma$ . Eq.  compares the one-body density $\rho_1^{\sigma}(z)$ at the trap center with its maximum 