Spontaneously Broken Spacetime Symmetries and Many-Body Phenomena

In this thesis, I discuss the application of Effective Field Theory (EFT) methods to a variety of problems across condensed matter and particle physics. Understanding how symmetries are realized by the system can put strong constraints on the low energy EFT which then dictates the macroscopic behavior of the system. We begin by utilizing these techniques to predict the leading order Landau parameter in the normal state of a strongly interacting cold-atomic fermi gas. We then use methods of coset construction to study magnon-phonon interactions in magnetic insulators where we are able to calculate the magnon lifetime as well predict interesting phenomena that can arise from explicit symmetry breaking effects. We also utilize the EFT in the context of dark matter detection to calculate one-magnon and multi-magnon emission rates for dark matter scattering. Finally, we study the dynamical modes of dislocations in solids and make predictions about the dispersion of these modes based on the lattice structure.