• electron-electron interactions
• Luttinger semimetals
• plasmons
• screening

Several two- and three-dimensional materials are known to present a gapless band structure, in which conduction and valence bands cross at the intrinsic Fermi level. Notable examples are graphene and Weyl semimetals, which present a linear Dirac-like dispersion in vicinity of their band touching points. Examples exist also of gapless materials exhibiting a contact point between quadratic valence and conduction bands. A parabolic node appears, for instance, in graphene bilayers in the AB stacking configuration and in three-dimensional materials such as mercury telluride (HgTe) and grey tin (αSn). Although the existence of quadratic gapless semimetals in three dimensions has been known for a long time, these materials continue to attract extensive attention because of the rich phenomenology stemming from their soft gapless bands.

This Thesis addresses many-body effects in three-dimensional parabolic semimetals in the weak-coupling Fermi-liquid regime, which arises when the Fermi level is away from the gapless point. The system is studied with the low-energy spin 3/2 Luttinger Hamiltonian, which stems from the constraints imposed by crystal symmetries, in the presence of long range Coulomb interactions. In particular, this Thesis addresses the evaluation of the linear density-density response function. Its knowledge allows to describe collective plasmon excitations, screening, and effective inter-particle interactions. The main original result of this Thesis is the analytical calculation of the density response function within the Random Phase Approximation. The dynamical dielectric function is then applied to the determination of the plasmon dispersion relation and to calculate the DC mobility of the many-body system in the presence of screened Coulomb impurities.