Existence of “$d$-wave” Pairs and Density Waves in a Class of Microscopic Models for High Transition Temperature Superconductors
High-temperature superconductors have different properties than conventional superconductors, one of these important properties is non-isotropic symmetry of the order parameter. In this work we present a model that shows the presence of symmetry $d$-wave in a class of superconductors of high $T_c$, as well as the condition for the existence of density waves in the states of equilibrium. We first analyze a three-body system, a bipolaron and two electrons, in order to study the effect of local and non-local Coulomb repulsion on the symmetry of the order parameter for the exact case and also the strong coupling limit. It is also shown the construction of an effective Hamiltonian that decouples the electron-bipolaron interaction in order to approach the collective problem of infinite bodies and we see how the proposed model predicts the formation of density waves for a specific region of the physical parameters of the studied Hamiltonian .