We analyze the low-energy excitations of diluted Josephson-junctions arrays, XY-quantum antiferromagnets, and interacting bosons near the percolation threshold. We then show that the
critical behavior of  these systems is dramatically altered by the presence of a Berry phase that leads to an order parameter precession and, depending on the system, is caused by an
external magnetic field or an external gate voltage. Most interestingly, the low energy excitations become spinless fermions with a 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.[1]

[1] Rafael M. Fernandes, Jörg Schmalian, Phys.Rev.Lett.106, 067004 (2011)