## 3 Results and Discussion  
### Magnetic Shell Structure and Phase Separation in the Trap  
Figure 2 displays the R-DMFT calculations of the particle density and polarization in the Lieb lattices of the \(30\times 30\) supercell structure confined in the trap potential of strength \(V_{0}=0.01\). The chemical potential is fixed at \(\mu_{\uparrow}=\mu_{\downarrow}=1\), which would give the half filling if the trap potential  
Figure 1: Lieb lattice structure and dynamical mean-field theory (DMFT) results of the magnetic phase transition at finite temperature. (**a**) The unit cell of the Lieb lattices (dashed lines) includes three sublattices, the site of the square lattices, and the two interstitial sites. (**b**) DMFT calculations of \(\langle S_{z}\rangle\) per unit cell in the translation invariant Lieb lattices at half-filling without considering a trap potential. The dotted line at \(T=0.05\) indicates the temperature used in the real-space dynamical mean-field theory (R-DMFT) calculations performed in the system confined in the trap. is absent. In the presence of the trap, we find that there exists a shell structure in the density and polarization profiles where the polarized core at the trap center is surrounded by the unpolarized outer shell in the trap edge. The trap used in Figure 2 contains \(N\approx 900\) particles with the total polarization \(P=(N_{\uparrow}-N_{\downarrow})/(N_{\uparrow}+N_{\downarrow})\approx 0.07\). Figure 2a shows the up(down)-spin-majority sites in the red(blue) color, and the spin-density-balanced sites in the gray. The size of the marker scales with the filling factor of the particles at the lattice site, indicating that the core shell has an almost uniform density of particles, while the density in the outer shell decays as it goes to the trap edge. The profiles of the density \(n\equiv n_{\uparrow}+n_{\downarrow}\) and magnetization \(m\equiv n_{\uparrow}-n_{\downarrow}\) along the line of \(y=0\) across the trap characterize the different phases of the core and outer shells. In the density profile shown in Figure 2b, the density plateau at \(n=1\) defines the core region at the trap center, whose incompressibility will be further examined below. In the surrounding outer shell, the densities are quite different between the sublattices A and B/C, and the sublattice B/C also shows an abrupt change in the density at the boundary between the core and surrounding outer shells. On the other hand, in the magnetization profile, the ferrimagnetic order appears in the core region, which is in good agreement with the previous DMFT result at the half-filling in the absence of the trap [53]. The sign of the magnetization \(m(x,y=0)\) spatially alternates between positive (up-spin majority) and negative (down-spin majority) along the sublattices A and B/C, as shown in Figure 2c. While this staggered magnetization is similar to the antiferromagnetism, it is, in fact, a ferromagnet since there are more up-spin particles than down-spin particles in the core. This magnetization profile also shows the plateau in the core region, which is abruptly dropped to zero as it enters into the region of the outer normal metal shell of the trap edge. Figure 2: Density and magnetization profiles of the Lieb-Lattice Hubbard model confined in the harmonic trap calculated by using R-DMFT. **(a)** Magnetization \(m(x,y)\equiv n_{\uparrow}(x,y)-n_{\downarrow}(x,y)\) visualized with the color code (red: up-spin majority, blue: down-spin majority, gray: zero magnetization) across the lattice sites in the trap. The circle with the dotted line indicates the size of the magnetic core \(R_{c}\) estimated based on the local density approximation. The density **(b)** and magnetization **(c)** profiles of the sublattices A and B plotted along the line of \(y=0\). The core region between \(-R_{c}\) and \(R_{c}\) is shaded in gray. The profiles for the sublattice C along \(x=0\) is the same as those for \(B\) along \(y=0\).