Polarizability of finite-bandwidth two-dimensional electron gas

The two-dimensional electron gas (2DEG), characterized by its parabolic electron energy dispersion, has been the subject of intense research for more than half a century. The main fuel to this interest has been its simplicity and physical relevance to describe the low energy behavior of most known 2D electronic systems, up to a point where any deviation from a 2DEG behavior (such as what was observed in graphene) has been pinpointed as anomalous and receives extra attention. Following this, here, we ask what would happen to the polarizability if we restrict the parabolic energy dispersion to a finite bandwidth (FBW), for which the 2DEG behavior would then be a natural limit (i.e. allowing for the bandwidth to be infinite). We discuss the polarizability of the FBW-2DEG, in dynamical and static limits, as the key physical quantity to obtain elementary excitation spectra, collective modes, charged impurity-limited transport properties, and spin-spin exchange interaction.