Comments on Extended t - J Models , Nodal Liquids and Supersymmetry ⋄

In the context of extended t − J models, with intersite Coulomb interactions of the form −V i,j n i n j , with n i denoting the electron number operator at site i, nodal liquids are discussed. We use the spin-charge separation ansatz as applied to the nodes of a d-wave superconducting gap. Such a situation may be of relevance to the physics of high-temperature superconductivity. We point out the possibility that at certain points of the parameter space supersymmetric points may occur, characterized by dynamical supersymmetries between the spinon and holon degrees of freedom, which are quite different from the symmetries in conventional supersymmetric t − J models. Such symmetries pertain to the continuum effective field theory of the nodal liquid, and one's hope is that the ancestor lattice model may differ from the continuum theory only by renormalization-group irrelevant operators in the infrared. We give plausible arguments that nodal liquids at such supersymmetric points are characterized by superconductivity of Kosterlitz-Thouless type. The fact that quantum fluctuations around such points can be studied in a controlled way, probably makes such systems of special importance for an eventual non-perturbative understanding of the complex phase diagram of the associated high-temperature superconducting materials.