Extended bound states and resonances of two fermions on a periodic lattice

The high-${\mathrm{T}}_{\mathrm{c}}$ cuprates are possible candidates for d-wave superconductivity, with the Cooper pair wave function belonging to a nontrivial irreducible representation of the lattice point group. We argue that this d-wave symmetry is related to a special form of the fermionic kinetic energy and does not require any novel pairing mechanism. In this context, we present a detailed study of the bound states and resonances formed by two lattice fermions interacting via a nonretarded potential that is attractive for nearest neighbors but repulsive for other relative positions. In the case of strong binding, a pair formed by fermions on adjacent lattice sites can have a small effective mass, thereby implying a high condensation temperature. For a weakly bound state, a pair with nontrivial symmetry tends to be smaller in size than an s-wave pair. These and other findings are discussed in connection with the properties of high-${\mathrm{T}}_{\mathrm{c}}$ cuprate superconductors.