Complex critical exponents for percolation transitions in Josephson-junction arrays, antiferromagnets, and interacting bosons.

We show that the critical behavior of the XY quantum-rotor model undergoing a percolation transition is dramatically affected by its topological Berry phase 2π ρ. In particular, for irrational ρ, its low-energy excitations emerge as spinless fermions with fractal spectrum. As a result, critical properties not captured by the usual Ginzburg-Landau-Wilson description of phase transitions arise, such as complex critical exponents, log-periodic oscillations and dynamically broken scale invariance.