  The S=1/2 Heisenberg model is considered on bilayer and single-layer square
lattices with couplings J1, J2, and with each spin belonging to one J2-coupled
dimer. A transition from a Neel to disordered ground state occurs at a critical
value of g=J2/J1. The systems are here studied at their dimer-dilution
percolation points p*. The multi-critical point (g*,p*) previously found for
the bilayer is not reproduced for the single layer. Instead, there is line of
critical points (g < g*,p*) with continuously varying exponents. The uniform
magnetic susceptibility diverges as 1/T^a $ with a in the range [1/2,1]. This
unusual behavior is attributed to an effective free-moment density T^(1-a). The
susceptibility of the bilayer is not divergent but exhibits remarkably robust
quantum-critical scaling.
