Lifshitz Tails of Scale-Invariant Theories with Electric Impurities

We study scale-invariant systems in the presence of Gaussian quenched electric disorder, focusing on the tails of the energy spectra induced by disorder. For relevant disorder we derive asymptotic expressions for the densities of unit-charged states in the tails, positing the existence of saddle points in appropriate disorder integrals. The resultant scalings are dictated by spatial dimensions and dynamical exponents of the systems.