Complex ground states of vortices and filaments

This dissertation consists of two parts. In the first part, we studied the ground state configurations of vortices with multi-scale inter-vortex interactions in layered superconductors. We found that by tuning the multi-scale interaction length, we could create vortex lattice ground states with different symmetries. It has been proposed that these structures can trap ultra-cold atoms for use in quantum emulators. In further work, we measured the phase diagram and discovered many new phases by changing the relative magnitude of the interaction ranges. In the second part, we analyzed the ground state configurations of confined filaments with long-range repulsive interactions. We first studied only straight filaments, using continuum limit analysis and numerical simulations, and we developed a new microscopic theory of defects. We found that in our system the gradient of the nonuniform density distribution can be controlled not only by the nature of the confinement but also by the range of the repulsive interactions. We also found that in this system, positive defects are much more difficult to create than negative defects. Furthermore, we showed that by using the local isotropic assumption, one can derive the relationship between nonuniform density and topological defects. In further work, we added a twist to the filament bundles. Generally speaking, uniform filament bundles do not form stable twist states. However, things change when we add defects into the center of the system. The filament bundles with negative disclinations will never form stable twist states, but there is a stable twisted filament bundle when there are positive disclinations. Furthermore, we generalized the local isotropic assumption to the twisted filament system and found a relationship between the nonuniform density, topological defects, and Gaussian curvature