Strong correlations in low dimensional conductors. What are they, and where are the challenges?

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 of 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.