• electron liquid
• flat band
• jellium

Flat band systems are metallic materials in which the band energy of the conduction band is only very weakly dependent on the (crystal) momentum, so that the kinetic energy can be considered as suppressed. Therefore, their properties are expected to be dominated by electron-electron interactions and thus to provide a fruitful playground for the exploration of strongly interacting phases. A well-known realization of a flat band system is the two-dimensional electron gas immersed in a magnetic field perpendicular to the plane of confinement, but with the advent of graphene other two realizations have become available, the rhombohedral stacked graphene \cite{PhysRevB.84.165404} and the twisted bilayer graphene \cite{BistritzerMacDonald}, in which the flat bands exist without the need of external fields. The twisted bilayer graphene has recently attracted a great deal of attention after the discovery \cite{UnconventionalSc,Correlated} of a superconducting transition observed within a phase diagram with the typical attributes of a strongly correlated system.
\par Motivated by these recent experimental findings, in this Thesis we have built a class of toy models with a flat band (TMFB), that generalize the rigid jellium model by letting the kinetic energy be proportional to a generic power $\alpha$ of the momentum, similarly to what happens in the rhombohedral stacked graphene. This family of models, which includes the standard jellium one for $\alpha=2$, features a single isotropic band, whose low-momentum region becomes essentially flat for high values of $\alpha$. The purpose of this work is to provide a basic characterization of the TMFB for high values of $\alpha$, computing the dynamic and thermodynamic properties of the ground state in the weakly interacting Fermi liquid regime.
\par The structure of the Thesis is as follows. The first Chapter is devoted to the description of the fundamental prerequisites for the subsequent original work. Firstly, the standard rigid jellium model \cite{GiuVi, Coleman}, which is the reference theory of the electron liquid, will be introduced, focusing on its many-body physics while completely disregarding the underlying lattice. The principal observables (energy, compressibility, response functions) will be explained, along with the basic tools for computing them in the weak coupling (i.e. high density) regime. A special emphasis will be placed on the random phase approximation (RPA) \cite{GiuVi, Coleman} and its improvements, which are some of the most important approximations in the many-body theory and will provide a basis for all the subsequent computations. Finally, the major theoretical, numerical and experimental results on graphene-based flat band systems will be presented.
\par After this introduction, the second Chapter develops the first part of the original results, namely the chief quantities of the noninteracting part of the toy model, and especially the numerical computation of the density response function (or polarization function) $\chi_{0}$, both in two and in three spatial dimensions. The imaginary part of the polarization function at real frequencies, which quantifies the dissipative properties of the system, and the full response function at imaginary frequencies (which is instead relevant in perturbative many-body computations) will be displayed and compared to the ordinary jellium case. We will show that, as the band flattens, the response accordingly shifts from high momenta (above the diameter of the Fermi sphere, $2 k_{\textup{F}}$) to lower ones and higher frequencies. This numerical work will be corroborated deriving some exact asymptotic properties of $\chi_{0}$.
\par In the final Chapter the interacting properties of the toy model will be investigated by numerical and analytical means, both in the two- and three-dimensional cases, mostly within the RPA. After a preliminary account on the estimated relevance of interactions of the TMFB, the numerical results for the correlation and the total energy, as well as the compressibility, will be presented and compared to the ordinary jellium case, observing a general trend of enhancement. Then, the RPA effective interaction will be discussed, and the numerically computed static effective potential will be shown to exhibit interesting $\alpha$-dependent attractive regions which may pave the way for intrinsic superconductivity. Eventually, the momentum dispersion of the plasmons in the TMFB will be computed analytically in RPA and compared to the $\alpha=2$ case, and the first corrections beyond RPA will be computed approximately.
\par This Thesis work is preliminary to the computation of the quasiparticle properties of the toy model and, possibly, to the exploration of broken-symmetry phases of flat band systems or to the generalization of the TMFB to two particle-hole symmetric bands.