Perturbing critical phenomena: from CPT to Holography

In this thesis we analyze different aspects of conformal field theories, one of the most powerful tools to study the physics of critical phenomena. Even though many progresses have been made in the knowledge and comprehension of conformal field theories, where the strong constraints imposed by conformal symmetry make the system easier to solve, still much needs to be done when a perturbation is introduced. We describe two different approaches which can be useful to tackle this problem. In particular in Chapter 2, we carefully describe the basic properties of Conformal Perturbation Theory (CPT), a mathematical tool to study conformal systems perturbed by relevant operators. We also provide two examples of how it can be applied and we match our finding with numerical Monte Carlo simulations. In Chapter 3, we study a conformal and relativistic hydrodynamic system,also making use of the AdS/CFT (or holographic) correspondence. This powerful technique, introduced in the context of string theory, becomes extremely relevant when one needs to study strongly correlated systems. In this case indeed, standard perturbative approaches usually fail: the holographic duality can be thought as a valid and complementary approach to investigate these important systems.