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The simplest way to add interactions to a single-band model of spinful electrons hopping on a crystal lattice is the Hubbard model, and the next simplest model is the extended Hubbard model (EHM). The EHM is useful for modeling superconductive electron pairing and emergent insulating phases including a dimerized phase called the "Bond Order Wave". In this thesis, I present two projects. In the first, I applied the numerical method of Density Matrix Renormalization Group to study the entanglement entropy at critical points of the clean EHM in one dimension near the BOW phase. In the second project, I applied analytical renormalization group approaches to study the phase diagram of the one dimensional model with weak disorder at weak coupling.

Direct drive inertial fusion energy (IFE) requires the injection, tracking, and engagement (illumination with high-powered lasers) of reflective spherical shells (targets) in order to produce fusion ignition and energy gain. Targets need to be tracked with 10-μm precision for this method of IFE to succeed. In this paper, one method for tracking targets is investigated, including a brief overview of existing tracking technology, analytical investigation of precision and design, and the results of a small-scale laboratory demonstration. This homodyne displacement measuring interferometer technique is labeled "fringe counting," and although the laboratory demonstration had mixed results, fringe counting may be capable of providing 10-micron precision measurements of target motion along the direction of travel for IFE. Robustness of the system is a serious concern. Suggestions for future improvement are provided. The conclusion includes an initial design for a full scale IFE fringe count system that incorporates many of the suggested improvements from this thesis. In addition to the main body material, several appendices are included. One examines potential for larger-scale injection/tracking demonstrations; another examines the Cat's eye retro- reflector and its invaluable use as an alignment tool for interferometers. And finally, a heterodyne system based on the concept of Zeeman splitting is investigated