This paper is written as a brief introduction for beginning graduate students. The picture of electron waves moving in a cristalline potential and interacting weakly with each other and with cristalline vibrations suffices to explain the properties o
f technologically important materials such as semiconductors and also simple metals that become superconductors. In magnetic materials, the relevant picture is that of electrons that are completely localized, spin being left as the only relevant degree of freedom. A number of recently discovered materials with unusual properties do not fit in any of these two limiting cases. These challenging materials are generally very anisotropic, either quasi one-dimensional or quasi two-dimensional, and in addition their electrons interact strongly but not enough to be completely localized. High temperature superconductors and certain organic conductors fall in the latter category. This paper discusses how the effect of low dimension leads to new paradigms in the one-dimensional case (Luttinger liquids, spin-charge separation), and indicates some of the attempts that are being undertaken to develop, concurrently, new methodology and new concepts for the quasi-two-dimensional case, especially relevant to high-temperature superconductors.
We apply the finite-temperature renormalization-group (RG) to a model based on an effective action with a short-range repulsive interaction and a rotation invariant Fermi surface. The basic quantities of Fermi liquid theory, the Landau function and t
he scattering vertex, are calculated as fixed points of the RG flow in terms of the effective actions interaction function. The classic derivations of Fermi liquid theory, which apply the Bethe-Salpeter equation and amount to summing direct particle-hole ladder diagrams, neglect the zero-angle singularity in the exchange particle-hole loop. As a consequence, the antisymmetry of the forward scattering vertex is not guaranteed and the amplitude sum rule must be imposed by hand on the components of the Landau function. We show that the strong interference of the direct and exchange processes of particle-hole scattering near zero angle invalidates the ladder approximation in this region, resulting in temperature-dependent narrow-angle anomalies in the Landau function and scattering vertex. In this RG approach the Pauli principle is automatically satisfied. The consequences of the RG corrections on Fermi liquid theory are discussed. In particular, we show that the amplitude sum rule is not valid.
