## 3 Results and Discussion
### Compressibility Tests of the Core by Adding Particles and Increasing Polarization  
In the sense of LDA, the density plateau with \(n=1\) of the core shell indicates that \(\frac{\partial n}{\partial t}=0\), suggesting zero compressibility. Energetically, this is due to the spectral gap [53] within which the change of chemical potential does not induce the change of the density. Thus, putting a particle into the trap, it is energetically difficult to go into the core shell in the trap center. At best, it would go to the boundary of the outer metallic shell and might participate in the core from the outermost part if it is energetically favored than being in the normal metallic phase. By using the R-DMFT calculations with the explicit inclusion of the trap potential, we examine more directly the effects of adding particles and increasing spin-population imbalance on the shell structure in the trap. Since R-DMFT is based on the formulation in the grand canonical ensemble, we control the spin-species-averaged chemical potential \(\mu_{0}\) and the Zeeman field \(h\) that determines the spin-dependent chemical potentials, \(\mu_{\uparrow}=\mu_{0}+h\) and \(\mu_{\downarrow}=\mu_{0}-h\), to add more particles and create the higher spin-population imbalance. Figure 4 presents the effects of increasing \(\mu_{0}\) at \(h=0\) on the density and magnetization profiles along the axis across the trap center. While the core area expands with more particles being added, the density plateau of \(n=1\) does not change in the core shell. This indicates that the added particles do not go to the interior of the core to raise the density at the core center, suggesting that the ferrimagnetic core is not compressible. The added particles go to the shell boundary and then enlarge the ferrimagnetic core from the outermost region. As the core area at constant density \(n=1\) gets larger with added particles, the spin population imbalance in the entire system also increases because the ferrimagnetic phase of the core has the fixed magnetization per site and per unit cell. Increasing \(\mu_{0}\) with \(h=0\) automatically leads to the higher polarization, accordingly, as the number of particles gets increased in the core. The next question would be, what would happen if we changed \(P\) to make the spin-population imbalance even larger than the total magnetization, that can fit in the core without breaking the ferrimagnetic phase? While it is numerically very time-consuming to tune \(\mu_{0}\) and \(h\) to fix the total particle number and spin polarization separately, here we simply increase the Zeeman field \(h\) at a fixed \(\mu_{0}\) to increase the polarization, which, in this case, accompanies the increase of the particle number as well. It turns out that the excess polarization goes to the metallic surrounding to make the outer shell polarized. Figure 5 presents the effects of increasing the Zeeman field \(h\) with the average chemical potential being fixed at \(\mu_{0}=1\). We observe that the ferrimagnetic core expands with increasing \(h\), and the surrounding outer shell gets polarized at a finite \(h\) since it is in the paramagnetic metallic phase. In the tests with the several values of \(h\), it turns out that the core of the ferrimagnetic phase is quite robust against the increase of the Zeeman field, implying that it is magnetically incompressible as well. Thus, adding spin-polarized particles of a particular spin species would not go inside of the core to increase the local density and magnetization at the trap center. Finally, we discuss the feature of the phase transition appearing in this trapped system and its sublattice-type dependence. In a harmonic trap, the phase of the quantum state depends on the position if the phase transition is driven by changing the density of particles or the local chemical potential in the sense of the LDA.