  Randomly diluted quantum boson and spin models in two dimensions combine the
physics of classical percolation with the well-known dimensionality dependence
of ordering in quantum lattice models. This combination is rather subtle for
models that order in two dimensions but have no true order in one dimension, as
the percolation cluster near threshold is a fractal of dimension between 1 and
2: two experimentally relevant examples are the O(2) quantum rotor and the
Heisenberg antiferromagnet. We study two analytic descriptions of the O(2)
quantum rotor near the percolation threshold. First a spin-wave expansion is
shown to predict long-ranged order, but there are statistically rare points on
the cluster that violate the standard assumptions of spin-wave theory. A
real-space renormalization group (RSRG) approach is then used to understand how
these rare points modify ordering of the O(2) rotor. A new class of fixed
points of the RSRG equations for disordered 1D bosons is identified and shown
to support the existence of long-range order on the percolation backbone in two
dimensions. These results are relevant to experiments on bosons in optical
lattices and superconducting arrays, and also (qualitatively) for the diluted
Heisenberg antiferromagnet.
