In this thesis, we have analyzed one-dimensional fermion-spin interactions in the dilute limit. The two cases we analyze represent different paradigms. For the first part, we look at the existence of spins for all sites as an effective model to describe the rearrangement of core electrons within the dynamic Hubbard model. Within this model, the behavior of electrons and holes will be compared in the presence of fermion-spin coupling and on-site repulsion. It will be shown that in this framework, electrons and holes behave differently and even though electrons experience increased repulsion, holes show attraction for a range of on-site repulsions. The characteristics of the interaction show effective nearest-neighbor attraction though no such term exists within the model. By the analysis of dynamic properties, two regions of interaction are identified. The gradual change from weak to strong coupling of fermions is presented. The effect of introducing on-site repulsion for both ranges of coupling is presented for both the dynamic Hubbard model and electron-hole symmetric version. For the second case involving fermion-spin interaction, we look at the interaction of a fermion with spins existing only for a small portion of the lattice, representing a coupled magnetic layer that an itinerant fermion interacts with through Heisenberg-like spin flip interaction. The interaction represents a spin-flip interaction of a spin current and magnetic layer. This interaction has been extensively studied for its relevance to computer hard drives both experimentally and theoretically. Most theoretical descriptions utilize the semi-classical Landau-Lifshitz-Gilbert (LLG) formalism. However, with recent improvements in experimental methods with very small magnetic layers and very fast real time measurements, quantum effects become more pronounced. We present quantum mechanical results that show considerable modification to spin-flip interaction. We identify a set of conditions that exhibits the existence of an emerging bound state for the spin current both numerically and analytically. The bound state is a quantum mechanical state and cannot be achieved with a classical picture. We present results in a one-dimensional lattice for a spin-1/2 system, and generalize our arguments to higher dimension and spins with S > 1/2.

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