Asymptotic safety in QFT : from quantum gravity to graphene

In this work we investigate properties of scale invariant theories. This kind of theories describe a variety of phenomena, and two particular examples are discussed. On the one hand, more than 100 years after the discovery of General Relativity by Einstein, we still don't know how to unify gravity and quantum mechanics. One possibility is that on very small scales, gravity could be scale invariant, allowing for a finite ultraviolet completion. On the other hand, we will study the phase diagram of graphene and related materials. Scale invariant points in phase diagrams are related to second order phase transitions, and near these universal behaviour is found. To investigate these systems, nonperturbative renormalisation group methods are used. In order to achieve trustworthy results, also technical progress, both analytical and numerical, had to be made. On the analytical side, the Mathematica package xAct is used to derive the equations underlying the scale invariance of the theories. To solve these numerically, pseudo-spectral methods are systematically introduced in the present context for the first time. The results thus obtained support the ultraviolet completion of gravity by a scale invariant point. The dependence on gauge fixing and parametrisation is investigated, and found to be reasonably small. The 2-loop counterterm, being the hallmark of the perturbative nonrenormalisability of gravity, is shown to be irrelevant at the scale invariant point. Finally, the split-Ward identities are partially solved by resolving correlation functions. Regarding graphene and similar materials, different levels of approximation show a very good convergence of results for critical exponents and anomalous dimensions at the phase transition studied. The combined power of both analytical and numerical methods excels particularly - without either, the calculations wouldn't be possible.