\
    E_{_{0,M}}^{MF}-V_{M}^{eff}(0) < 0 
    \ \land \
    E_{_{0,I}}^{MF}-V_{I}^{eff}(0) > 0  \\
    \text{SB} &\mbox{:} \
    d > 0
    \end{cases}$$ where $E_{_{0,\sigma}}^{MF}$ is the ground state energy of . As long as the ground state energy of the effective species Hamiltonian exceeds the effective potential height at the trap center, the species remains at the trap center. Phase diagrams produced this way match exactly the ones shown in Fig. \[fig:MF\_few\_body.eps\].

The interpretation is now as follows. For a very weak coupling both the majority and the impurity reside in the ground state of the harmonic oscillator. Once the induced potential $V_{\sigma}^{ind}$ of species $\sigma$ becomes large enough to produce a barrier in $V_{\sigma}^{eff}$, the corresponding density $\rho_{_{MF}}^{\sigma}$ will start to expand. By growing in width it will prevent the other component $\bar{\sigma}$ from developing a barrier of its own. When the height of the potential barrier becomes of the same magnitude as the lowest energy of the corresponding effective potential, the species $\sigma$ splits into two fragments. Then it starts squeezing the other component $\bar{\sigma}$ by increasing the effective trap frequency of the renormalized harmonic oscillator $V_{\bar{\sigma}}^{eff}$.

The barrier in $V_{\sigma}^{eff}$ appears once the following condition is fulfilled: $$\label{eq:barrier_criterion}
    \exists \ x_0 