$d_{x^2-y^2}$ superconductivity in a model of correlated fermions

A two-dimensional fermionic model with attractive interactions is discussed. Its form is motivated by the phenomenology of the underdoped high-$T_c$ cuprates, as discussed in previous literature. The exact solution to the two-particle problem leads to a bound state in the $d_{x^2-y^2}$ subspace. Numerical techniques show that the model has strong $d_{x^2-y^2}$ pairing correlations in the ground state at low fermionic density. Within a self-consistent random phase approximation diagrammatic study, the model has a $d$-wave superconducting (SC) phase. The density dependence of the critical temperature is calculated. We argue that in the context of $d$-wave SC this model fulfills the role that the attractive on-site Hubbard model has played for $s$-wave SC. We also show that another candidate, the attractive $t-U-V$ model, is actually not useful as a toy model for $d$-wave SC for a variety of reasons.