the ground state and the first excited state of the many-body spectrum, which becomes an exact degeneracy in the limit of $g_{_{MI}} \to \infty$. Once this limit is reached, any superposition of those two states is also an eigenstate of . Since they are of different parity symmetry $P$ and $[H,P]=0$, it is possible to choose the superposition to be parity symmetric or to break the parity symmetry of . It was suggested [@phasesZollner2008] that the corresponding many-body wave-function may be written in terms of number states as $\ket{\Psi} =c_1 \ket{N_M}_L \ket{0_M}_R \otimes \ket{0_I}_L \ket{N_I}_R
+ c_2 \ket{0_M}_L \ket{N_M}_R \otimes \ket{N_I}_L \ket{0_I}_R$ with two parity-broken SPFs $\varphi^{\sigma}_j(z)$ featuring an asymmetric Gaussian shape with a maximum on the j=L(eft) or j=R(ight) side w.r.t. the trap center. Within the mean-field approximation the eigenenergy of the first excited state coincides with the ground state energy already for a finite coupling $g_{_{MI}}$. Since mean-field does not incorporate entanglement, the state collapses either to $\ket{N_M}_L \otimes \ket{N_I}_R$ or to $\ket{N_M}_R \otimes \ket{N_I}_L$, resulting in a phase with broken parity symmetry.

With this we end our overview over different phases and showcase a compact summary of the phases: $$\label{eq:classification}
    \begin{cases} 
    \text{M} & \mbox{:} \
    d = 