Crystals of topological solitons. An effective field theory approach

The number of constituents that compose macroscopic systems makes condensed matter a very diverse area of physics. Therefore, it is of extreme importance to discover common themes and properties that permit the description of distinct systems similarly. One of these themes is to identify and study gapless excitations at long wavelengths and small energies. Crystals are the most familiar systems that exhibit gapless degrees of freedom, called phonons. This thesis applies field theory methods to study long-wavelength excitations of topological solitons that arrange themselves into crystal structures and emerge as ground states of superfluids and chiral magnets. First, using boson-vortex duality, we formulate a low-energy effective theory of a two-dimensional vortex lattice in a bosonic Galilean-invariant compressible superfluid by describing vortices as two-dimensional point charges moving in a magnetic background. We extract the excitation spectrum, which contains a gapped Kohn mode and an elliptically polarized Tkachenko mode that has quadratic dispersion relation at low momenta. We couple the theory to an external U(1) gauge field and extract the particle number current transport. Second, we study edge waves travelling along the lattice's boundary in two-dimensional skyrmion lattices which appear in thin-film chiral magnets. In elastic systems, these excitations are known as Rayleigh waves. We find that the direction of propagation of the Rayleigh modes is determined not only by the thin film's chirality but also by the Poisson ratio of the crystal. Furthermore, we discover three qualitatively different regions distinguished by the low-frequency edge waves' chirality and inspect their properties.