Field theories in condensed matter physics

In this thesis, field theory is applied to different problems in the context of condensed matter physics. In the first part of this work, a classical problem in which an elastic instability appears is studied. By taking advantage of the symmetries of the system, it is shown that when a soft substrate has a stiff crust and the whole system is forced to reduce its volume, the stiff crust rearranges in a way that will break the initial rotational symmetry, producing a periodic pattern that can be manipulated at our will by suitable changes of the external parameters. It is shown that elastic interactions in this type of systems can be mapped into non-local effective potentials. The possible application of these instabilities is also discussed. In the second part of this work, quantum electrodynamics (QED) is analyzed as an emergent theory that allows us to describe the low energy excitations in two-dimensional nodal systems. In particular, the ballistic electronic transport in graphene-like systems is analyzed. We propose a novel way to control massless Dirac fermions in graphene and systems alike by controlling the group velocity through the sample. We have analyzed this problem by computing transport properties using the transmission matrix formalism and, remarkably, it is found that a behavior conforming with a Snell's-like law emerges in this system: the basic ingredient needed to produce electronic wave guides. Finally, an anisotropic and strongly interacting version of QED 3 is applied to explain the non-universal emergence of antiferromagnetic order in cuprate superconductors. It is pointed out that the dynamics of interacting vortex anti-vortex fluctuations play a crucial role in defining the strength of interactions in this system. As a consequence, we find that different phases (confined and deconfined) are possible as a function of the relative velocity of the photons with respect to the Fermi and gap velocities for low energy excitation in cuprates.