parity-symmetric humps with a local minimum at the trap center ($\Delta_{\sigma}<1$ and $d=0$) we have a composite fermionization phase $CF$. On the level of one-body densities $CF$ appears to be miscible owing to the high spatial overlap between the components. However, the deviations to the miscible phase become evident upon investigating the two-body density matrices and . Namely, two particles of the same component can be found either on the left or the right side w.r.t. trap center, while two particles of different components are always on opposite sides.

While the core-shell phases $IMI$ and $MIM$ do not rely on entanglement between the components, $CF$ is always an entangled many-body state made out of two major species orbitals $S=2$ and two major SPFs $s_{\sigma}=2$ on the particle layer. Thus, $CF$ cannot be obtained within the mean-field approximation. In fact, we observe that once the entanglement of the true many-body state, characterized by the von Neumann entropy (see Sec. \[sec:few\_body\_MLX\]), reaches a certain threshold, a collapse to a phase with broken parity symmetry ($d>0$) will take place in the mean-field picture. We abbreviate this phase with $SB$ from now on.

The origin of $SB$ is the onset of a quasi-degeneracy between 