Thesis advisor: Kevin S. Bedell / We present several results based upon non-trivial extensions of Landau-Fermi liquid theory. First proposed in the mid-20th century, the Fermi liquid approach assumes an adiabatic “switching-on” of the interaction, which allows one to describe the collective excitations of the many-body system in terms of weakly-interacting quasiparticles and quasiholes. At its core, Landau-Fermi liquid theory is often considered a perturbative approach to study the equilibrium thermodynamics and out-of-equilibrium response of weakly-correlated itinerant fermions, and therefore non-trivial extensions and consequences are usually overlooked in the contemporary literature. Instead, more emphasis is often placed on the breakdown of Fermi liquid theory, either due to strong correlations, quantum critical fluctuations, or dimensional constraints. After a brief introduction to the theory of a Fermi liquid, I will first apply the Landau quasiparticle paradigm to the theory of itinerant Majorana-like fermions. Defined as fermionic particles which are their own anti-particle, traditional Majorana zero modes found in topological materials lack a coherent number operator, and therefore do not support a Fermi liquid-like ground state. To remedy this, we will apply a combinatorical approach to build a statistical theory of self-conjugate particles, explicitly showing that, under this definition, a filled Fermi surface exists at zero temperature. Landau-Fermi liquid theory is then used to describe the interacting phase of these Majorana particles, from which we find unique signatures of zero sound in addition to exotic, non-analytic contributions to the specific heat. The latter is then exploited as a “smoking-gun” signature for Majorana-like excitations in the candidate Kitaev material Ag3LiIr2O6, where experimental measurements show good agreement with a sharply-defined, “Majorana-Fermi surface” predicted in the underlying combinatorial treatment. I will then depart from Fermi liquid theory proper to tackle the necessary conditions for the applicability of Luttinger’s theorem. In a nutshell, Luttinger’s theorem is a powerful theorem which states that the volume of phase space contained in the Fermi surface is invariant with respect to interaction strength. In this way, whereas Fermi liquid only describes fermionic excitations near the Fermi surface, Luttinger’s theorem describes the fermionic degrees of freedom throughout the entire Fermi sphere. We will show that Luttinger’s theorem remains valid only for certain frequency and momentum-dependencies of the self-energy, which correlate to the exis- tence of a generalized Fermi surface. In addition, we will show that the existence of a power-law Green’s function (a unique feature of “un-particle” systems and a proposed characteristic of the pseudo-gap phase of the cuprate superconductors) forces Luttinger’s theorem and Fermi liquid theory to be mutually exclusive for any non-trivial power of the Feynman propagator. Finally, we will return to Landau-Fermi liquid theory, and close with novel out-of-equilibrium behavior and stability in unconventional Fermi liquids. First, we will consider a perfectly two- dimensional Fermi liquid. Due to the reduction in dimension, the traditional mode expansion in terms of Legendre polynomials is modified to an expansion in terms of Chebyshev polynomials. The resulting orthogonality conditions greatly modifies the stability and collective modes in the 2D system. Second, we will look at a Fermi liquid in the presence of a non-trivial gauge field. The existence of a gauge field will effectively shift the Fermi surface in momentum space, resulting in, once again, a modified stability condition for the underlying Fermi liquid. Supplemented with a modernized version of Mermin’s condition for the propagation of zero sound, we outline the full effects a spin symmetric or anti-symmetric gauge would have on a Fermi liquid ground state. / Thesis (PhD) — Boston College, 2021. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.