Zero-point quantum fluctuations of a Néel order can produce effective interactions between quasi-orphan spins weakly coupled to the lattice. On the √3×√3-distorted triangular lattice, this phenomenon leads to a correlated partially disordered phase. In this article, we use matrix product state methods to study a similar model: the S=1/2 stuffed square lattice. Tuning the exchange amplitudes we go from a square lattice plus orphan central spins at J′/J=0, to the union jack lattice at J′/J=1, and a square lattice including all spins at J/J′=0. We calculate the complete antiferromagnetic phase diagram, dominated by ferrimagnetic and Néel orders, and compare it with existing results. Most importantly, we find a partially disordered phase in the weakly frustrated regime. In this phase, the Néel order from the square lattice is unaffected, while the central spins form a collective state with exponentially decaying double-striped correlations. We also study the role of quantum fluctuations by introducing an ordering staggered magnetic field on the square sublattice and find that the central spins order ferromagnetically when fluctuations from the Néel order are suppressed.
