• conformal field theory
• conformal perturbation theory
• Harris criterion
• numerical conformal bootstrap
• quenched disorder
• random systems
• renormalization group
• Replica Trick
• universality
• XY model

The presence of impurities and defects in real systems requires to extend the standard apparatus of universality and Renormalization Group. This thesis analyzes the effect of the introduction of a quenched (i.e. time independent) disorder in the critical O(2) vector model. We are interested in understanding whether the disorder modifies the critical behaviour of the pure system, that is a Conformal Field Theory. The so-called Harris criterion allows us to predict qualitatively the stability of the pure fixed point against disorder. In order to perform calculations, we use the Replica Trick to rewrite the average of the partition function over the disorder in a simple way. In doing this, we obtain a set of (n) identical independent copies of the initial system, each of them coupled to the same disorder. However, the Path Integral in the impurities leads to an interaction term between the replicas. We analyze the evolution of this coupling with the tools of Conformal Perturbation Theory: knowing the CFT data, namely the correlators, of the pure theory, we deduce the properties of the random system studying the RG flow. We first consider the random temperature perturbation, corresponding to insert the singlet of O(2) with a random coupling. Our strategy is to perfom the analysis in d=4-ε and then attempt the computation directly in d=3. In this case we start from a strongly interacting CFT, that is why we use the results recently obtained with numerical conformal bootstrap to write down the beta-function for the disorder-induced coupling. This kind of disorder is found to be relevant close to 4d but irrelevant in 3d, requiring a future deeper investigation with resummation techniques. Finally we present the interesting case of a random perturbation associated with the lowest-dimension charge-2 scalar of the O(2) model. We expect to find a new IR fixed point since this operator should be relevant in d=3 from conformal bootstrap studies. Contrary to the random temperature case, this perturbation is charged under the O(2) group: the possible symmetry breaking depends on which component of the disorder-induced interaction is turned on.