  Using an adaptation of Schwinger boson mean field theory (SBMFT) for
nonuniform systems, we study the nature of low-energy spin excitations on the
square and Bethe lattice at their percolation threshold. The optimal SBMFT
parameters are interpreted as on-site potentials and pairing amplitudes, which
enables an explanation of why emergent local moments develop in this system on
dilution [L. Wang and A. W. Sandvik, Phys. Rev. Lett. 97, 117204 (2006); H.J.
Changlani et al., ibid 111, 157201 (2013)] and why the corresponding single
particle frequencies are driven to anomalously low values. We discuss how our
mean field calculations suggest the strong link between the presence of
sublattice imbalance and long range antiferromagnetic order, and why linear
spin wave theory is inadequate for capturing this relation. Within the SBMFT
framework, we also extract an energy scale for the interaction between emergent
moments, which show qualitative agreement with many-body calculations.
