Two different 2D Heisenberg S=1/2 antiferromagnets consisting of coupled
dimers have been studied using quantum Monte Carlo simulations; a bilayer
where a dimer consists of spins in different layers and and a single layer
with staggered dimers. Due to the tendency to singlet-formation on the
dimers, these models have quantum-critical points as a function of the
intra-bilayer coupling. By randomly removing a fraction p of the dimers,
quantum-criticality in the presence of disorder can be studied. Here the
focus will be on the behavior at the geometrical percolation point, where
the spin system on the percolating cluster has a fractal dimensionality
d=91/48. The critical coupling at which the percolating cluster becomes
quantum-critical is extracted and some critical exponents are calculated
as well. It is shown that the critical point is different for the two
models, likely due to the layer-exchange symmetry of the bilayer model.
The behavior at pecolation for coulings less than the critical coupling
is also different. In the bilayer model the dilution-driven transition in
this case is a classical percolation transition, whereas the single-layer
model exhibits a line of quantum-critical points with continuoulsy
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