In this work, we describe the simplifications to the Extremely Correlated Fermi Liquid Theory (ECFL) \cite{ECFL, Monster} which occur in the limit of infinite spatial dimensions. In particular, we show that the single-particle electron Green's function G(k) can be written in terms of two momentum-independent self-energies. Moreover, we elucidate the nature of the ECFL \lambda expansion in the limit of infinite dimensions and carry out this expansion explicitly to O(\lambda^2). Additionally, we demonstrate the vanishing of vertex corrections to the optical conductivity in general and to each order in \lambda in the limit of infinite dimensions. We generalize the ECFL formalism to the infinite-U Anderson impurity model (AIM) , and demonstrate a Dynamical Mean-Field Theory (DMFT) like mapping between the ECFL objects of the infinite-dimensional t-J model and the infinite-U AIM, and show that this mapping holds to each order in \lambda. We compute the spectral function for the AIM to O (\lambda^2) and make comparisons with results obtained through Numerical Renormalization Group (NRG) computations. Finally, we develop a novel formalism for the high-temperature expansion of dynamical correlation functions in the infinite-U Hubbard model which is more efficient than any used previously and gives results for an arbitrary number of spin species. We use it to calculate the single-particle Green's function G(k) to fourth order in (\beta t) for m spin-species on a d-dimensional hypercube.

In this work, we apply the extremely correlated Fermi liquid (ECFL) theory into studying the $t$-$t'$-$J$ model in one and two dimensions. In 1-d, we combine ECFL and time-dependent density matrix renormalization group (tDMRG) method. The two methods provide a unique insight into the strong momentum dependence of the self-energy of this prototypical non-Fermi liquid, described at low energies as a

Tomonaga-Luttinger liquid. We also demonstrate its intimate relationship to spin-charge separation,

i.e. the splitting of Landau quasiparticles of higher dimensions into two constituents, driven by

strong quantum fluctuations inherent in one dimension. In 2-d, low energy properties of the metallic state of the two-dimensional $t$-$J$ model are presented for second neighbor hopping with hole-doping ($t'\leq0$) and electron-doping ($t'>0$), with various superexchange energy $J$. The density and temperature dependent spectral properties, resistivity and Hall response are calculated. The spectral features display high thermal sensitivity at modest T for density $n\geq 0.8$, implying a suppression of the effective Fermi-liquid temperature by two orders of magnitude relative to the bare bandwidth. Flipping the sign of $t'$ or varying doping $\delta$ changes the curvature of the resistivity versus T curves between convex and concave, consistent to experimental findings.

In this dissertation, we present Extremely Correlated Fermi Liquid (ECFL) description of low energy excitation spectrum of high temperature cuprate superconductors measured by Angle Resolved Photoemission Spectroscopy (ARPES). Focusing on interpretation of the ARPES data, we propose a rigorous approach to understanding the unconventional quasiparticle dynamics in high Tc cuprate superconductors. First, we present ARPES line shapes fitting with the ECFL theory. The ECFL theory is very successful in explaining the ARPES spectral functions of high Tc cuprate superconductors, and it fits the ARPES line shapes as functions of momentum (momentum distribution curves, MDCs) and energy (energy distribution curves, EDCs) for different materials and different temperature with the same intrinsic physical variables along the nodal direction. Although it is not the first to fit both MDCs and EDCs of high Tc cuprate superconductors, the ECFL theory offers unprecedented applicability in fitting the ARPES spectral functions. Second, the ECFL theory provides a robust discussion for the origin of kinks in energy dispersion of strongly correlated material measured by ARPES. A bending anomaly in the energy dispersion of strongly correlated matter, the universal low energy kink in the ARPES spectrum ~ 50 - 100 meV, can not be explained within the standard linear band dispersion theory because of significant corrections due to interactions. In our work, we address correlation kinks arising from the momentum dependent Dyson self-energy of the ECFL theory. The calculation is overdetermined, and four independent variables can predict sets of measurable relations in the ARPES experiments. We find that ECFL interpretation of the ARPES kinks is consistent with the available experimental data, and we provide a decisive set of predictions for future high resolution ARPES experiments.

The primary aim of this thesis is to understand the effects of electronic frustration generated by local interactions of the Hubbard U type on a lattice. Specifically we attempt to study the strongly interacting liquid phase of the Hubbard model and the so called extremely correlated phase of the t-J model without any broken symmetries. Of particular interest is the fate of the Fermi Liquid state under the influence of a large U. Exact solutions in solvable limits inform most of the physical intuition. Perturbative methods such as an infinite order calculation in the particle-particle channel ladder expansion yield fruitful results. Finally the non-perturbative field theory of Schwinger is used to generate a calculation scheme for the Green's function of the non-canonical Fermions of the t-J model. Numerical results are compared with important experimental results from the high-T_c Cuprate compounds.