Studies in strongly coupled quantum field theories and renormalization group flows

This thesis presents studies in strongly coupled Renormalization Group (RG) flows. In the first part, we analyze the subject of non-local Conformal Field Theories (CFTs), arising as continuous phase transitions of statistical models with long-range interactions. Specifically, we study the critical long-range Ising model in a general number of dimension: first we show that it is conformally invariant, and then we study in depth the different regimes of the theory. We find an example of an infrared duality, to our knowledge the first non-local example of such phenomenon. The second part of the thesis deals with walking theories and weakly first order phase transi- tions, meaning Quantum Field Theories that show approximate scale invariance over a range of energies, in a general number of dimensions. We discuss several example in the high energy as well as the statistical mechanics literature, and show that these theories can be understood as an RG flow passing between two complex CFTs, i.e. non-unitary theories living at complex values of the couplings. Combining the conformal data of these complex CFTs and conformal perturbation theory, we describe observables of the walking theory. Finally, we give the explicit example of the two dimensional Potts model with more than four states.