Finite-temperature Lindhard function for α--T₃ materials

We have calculated the static and finite-temperature Lindhard function from a general polarization function Π_φ⁽⁰⁾(q, ω | T) in the static limit ω --> 0 for α--T₃ materials with an arbitrary hopping parameter 0 < α < 1. Our results are obtained in the form of semi-analytical expressions, i.e. one of the two required integrals has been performed analytically, leading to an analytical function. The other integral, however, could be computed explicitly in the limit of either low or high temperature, including the limit of T --> 0. Physically, the static-limit polarization function is a very important quantity for calculating Boltzmann transport, electron elastic scattering, and the static screening of electrons in any newly discovered two-dimensional material. Importantly, we have recovered our previously acquired analytical expressions for graphene and for a dice lattice as α --> 0 or α --> 1, respectively.