  We show that the critical behavior of quantum systems undergoing a
percolation transition is dramatically affected by their topological Berry
phase $2\pi\rho$. For irrational $\rho$, we demonstrate that the low-energy
excitations of diluted Josephson-junctions arrays, quantum antiferromagnets,
and interacting bosons are spinless fermions with fractal spectrum. As a
result, critical properties not captured by the usual Ginzburg-Landau-Wilson
description of phase transitions emerge, such as complex critical exponents,
log-periodic oscillations and dynamically broken scale-invariance.
