The usual virial theorem, relating kinetic and potential energy, is extended to a molecule by the use of the true wave function. The virial theorem is also obtained for a molecule from a trial wave function which is scaled separately for electronic and nuclear coordinates.

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Thesis advisor: Kevin S. Bedell / One of the many achievements of renowned physicist L.D. Landau was the formulation of Fermi Liquid Theory (FLT). Originally debuted in the 1950s, FLT has seen abundant success in understanding degenerate Fermi systems and is still used today when trying to understand the physics of a new interacting Fermi system. Of its many advantages, FLT excels in explaining why interacting Fermi systems behave like their non-interacting counterparts, and understanding transport phenomena without cumbersome and confusing mathematics. In this work, FLT is applied to systems whose low energy excitations obey the massless Dirac equation; i.e. the energy dispersion is linear in momentum, ε α ρ, as opposed to the normal quadratic, ε α ρ². Such behavior is seen in numerous, seemingly unrelated, materials including graphene, high T[subscript]c superconductors, Weyl semimetals, etc. While each of these materials possesses its own unique properties, it is their low energy behavior that provides the justification for their grouping into one family of materials called Dirac materials (DM). As will be shown, the linear spectrum and massless behavior leads to profound differences from the normal Fermi liquid behavior in both equilibrium and transport phenomena. For example, with mass having no meaning, we see the usual effective mass relation from FLT being replaced by an effective velocity ratio. Additionally, as FLT in d=2 has been poorly studied in the past, and since the most famous DM in graphene is a d=2 system, a thorough analysis of FLT in d=2 is presented. This reduced dimensionality leads to substantial differences including undamped collective modes and altered quasiparticle lifetime. In chapter 3, we apply the Virial theorem to DM and obtain an expression for the total average ground state energy $E=\frac{B}{r_s}$ where $B$ is a constant independent of density and $r_s$ is a dimensionless parameter related to the density of the system: the interparticle spacing $r$ is related to $r_s$ through $r=ar_s$ where $a$ is a characterstic length of the system (for example, in graphene, $a=1.42$ \AA). The expression derived for $E$ is unusual in that it's typically impossible to obtain a closed form for the energy with all interactions included. Additionally, the result allows for easy calculation of various thermodynamic quantities such as the compressibility and chemical potential. From there, we use the Fermi liquid results from the previous chapter and obtain an expression for $B$ in terms of constants and Fermi liquid parameters $F_0^s$ and $F_1^s$. When combined with experimental results for the compressibility, we find that the Fermi liquid parameters are density independent implying a unitary like behavior for DM. In chapter 4, we discuss the alleged universal KSS lower bound in DM. The bound, $\frac{\eta}{s}\geq\frac{\hbar}{4\pi k_B}$, was derived from high energy/string theory considerations and was conjectured to be obeyed by all quantum liquids regardless of density. The bound provides information on the interactions in the quantum liquid being studied and equality indicates a nearly perfect quantum fluid. Since its birth, the bound has been highly studied in various systems, mathematically broken, and poorly experimented on due to the difficult nature of measuring viscosity. First, we provide the first physical example of violation by showing $\frac{\eta}{s}\rightarrow 0$ as $T\rightarrow T_c$ in a unitary Fermi gas. Next, we determine the bound in DM in d=2,3 and show unusual behavior that isn't seen when the bound is calculated for normal Fermi systems. Finally we conclude in chapter 5 and discuss the outlook and other avenues to explore in DM. Specifically, it must be pointed out that the physics of what happens near charge neutrality in DM is still poorly understood. Our work in understanding the Fermi liquid state in DM is necessary in understanding DM as a whole. Such a task is crucial when we consider the potential in DM, experimentally, technologically, and purely for our understanding. / Thesis (PhD) — Boston College, 2020. / Submitted to: Boston College. Graduate School of Arts and Sciences. / Discipline: Physics.