We study the effects of quenched disorder and a dissipative Coulomb interaction on an anyon gas in a periodic potential undergoing a quantum phase transition. We use a (2+1)−dimensional low-energy effective description that involves Nf=1 Dirac fermion coupled to a U(1) Chern-Simons gauge field at level (θ−1/2). When θ=1/2 the anyons are free Dirac fermions that exhibit an integer quantum Hall transition; when θ=1 the anyons are bosons undergoing a superconductor-insulator transition in the universality class of the three-dimensional XY model. Using the large Nf approximation we perform a renormalization-group analysis. We find the Coulomb interaction to be an irrelevant perturbation of the clean fixed point for any θ. The dissipative Coulomb interaction allows for two classes of IR stable fixed points in the presence of disorder: those with a finite nonzero Coulomb coupling and dynamical critical exponent z=1 and those with an effectively infinite Coulomb coupling and 1<z<2. At θ=1/2 the clean fixed point is stable to charge-conjugation preserving (random mass) disorder, while a line of diffusive fixed points is obtained when the product of charge-conjugation and time-reversal symmetries is preserved. At θ=1 we find a finite disorder fixed point with unbroken charge-conjugation symmetry whether or not the Coulomb interaction is present. Other cases result in runaway flows. We comment on the relation of our results to other theoretical studies and the relevancy to experiment
