Complex Critical Exponents in Diluted Systems of Quantum Rotors

In this work, we investigate the effects of the Berry phase $2 \pi \rho$ on the critical properties of $XY$ quantum-rotors that undergo a percolation transition. This model describes a variety of randomly-diluted quantum systems, such as interacting bosons coupled to a particle reservoir, quantum planar antiferromagnets under a perpendicular magnetic field, and Josephson-junction arrays with an external bias-voltage. Focusing on the quantum critical point at the percolation threshold, we find that, for rational $\rho$, one recovers the power-law behavior with the same critical exponents as in the case with no Berry phase. However, for irrational $\rho$, the low-energy excitations change completely and are given by emergent spinless fermions with fractal spectrum. As a result, critical properties that cannot be described by the usual Ginzburg-Landau-Wilson theory of phase transitions emerge, such as complex critical exponents, log-periodic oscillations, and dynamically-broken scale invariance.