Despite great theoretical effort since the conception of manybody physics to elucidate

the nature of interacting fermions — in a precise and quantitative manner — the phase

diagrams of many correlated electronic systems remain impenetrable to even the most

rigorous and skilled approaches. In this manuscript I detail the new techniques, mod-

els, and solid-state materials that lie along the contemporary boundaries of scientific

understanding in strongly correlated materials. In particular, I discuss the fundamental

obstacles of manybody electronic systems and discuss how these issues are exemplified

by the various results reviewed. The work presented in Chapters 2-4 are representative

of my time at UCSB, and reflect the collaborations and publications of my Ph.D.