Gravity, holography and applications to condensed matter

Momentum relaxation is an ever-present and unavoidable ingredient of any realistic condensed matter system. In real-world materials the presence of a lattice, impurities or disorder forces momentum to dissipate and leads to relevant physical effects such as the finiteness of the DC transport properties, i.e. conductivities. The main purpose of this thesis is the introduction of momentum dissipation and its consequent effects into the framework of AdS/CMT, namely the applications of the gauge-gravity duality to condensed matter. A convenient and effective way of breaking the translational symmetry associated to such a conservation law is provided by massive gravity (MG) bulk theories. We consider generic massive gravity models embedded into asymptotically Anti de Sitter spacetime and we analyze them using holographic techniques. We study in detail their consistency and stability. We then focus our attention on the transport properties of the CFT duals. A big part of our work is devoted to the analysis of the electric conductivity in relation to possible universal bounds and the existence of holographic metal-insulator transitions. We moreover initiate the study of the viscoelastic response and we consider the possible violation of the well known KSS bound. We finally describe the effects of momentum relaxation on the well known holographic models for superconductivity.