order? Such problems were first addressed in Refs. and for the PAM and KLM, respectively, within a mean-field approach. Recent mean-field results for the PAM, presented in Ref., also provide evidence of a ferromagnetic ground state. Similarly, unbiased methods for an analogous spin model, namely the bilayer Heisenberg model (Ref.), shows that, indeed, the depletion of spins induces an AFM ground state. As we shall see in the next Section, magnetic long range order indeed occurs, even deep in what was previously the singlet phase of the undepleted model (large $d$-$f$ hybridization). However, as presented below, the system still has a “memory" of the old critical point.

Half depletion {#Half depletion}
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![(Color online) The lattice geometry for the regularly half depleted PAM. The unit cell is composed by the sites $d_{0}$, $d_{1}$ and $f$. Here we depict a lattice with $N_{cells} = 2 \times 2^{2}$.[]{data-label="Fig:Lattice_half"}](Lattice_PAM_depleted.pdf)

The previous results suggest that if the number of nonmagnetic defects increases, the magnetic correlations can be enhanced, and thus the ground state may exhibit magnetic long range order even if the undepleted model is in the singlet phase. Here we explore the case of depletion of half of the $f$-orbitals, in the checkerboard 