We show that a scale anomaly appears for massless Dirac fermions in a Coulomb potential. This example of quantum phase transition shows up in the low energy spectrum of quasi-bound states. 

By varying the electric charge, a sharp transition is observed between a single quasi-bound state and a discrete scale invariant Efimov like spectrum at a critical value which depends solely on the spatial dimension. 

This transition is directly and clearly observed in the STM analysed local density of states of charged single vacancies created in graphene layers. A quantitative analysis is presented based on a simple Dirac model which corroborates the experiment. 

Moreover, we find that the vacancy necessarily breaks parity in the Dirac description of graphene.  The universality of this quantum phase transition is demonstrated by comparing our results to the experimental Efimov spectrum determined using cold atomic systems.