  The t-J model is believed to be a minimal model that may be capable of
describing the low-energy physics of the cuprate superconductors. However,
although the t-J model is simple in appearance, obtaining a detailed
understanding of its phase diagram has proved to be challenging. We are
therefore motivated to study modifications to the t-J model such that its phase
diagram and mechanism for d-wave superconductivity can be understood
analytically without making uncontrolled approximations. The modified model we
consider is a $t'$-$J_z$-$V$ model on a square lattice, which has a
second-nearest-neighbor hopping $t'$ (instead of a nearest-neighbor hopping
$t$), an Ising (instead of Heisenberg) antiferromagnetic coupling $J_z$, and a
nearest-neighbor repulsion $V$. In a certain strongly interacting limit, the
ground state is an antiferromagnetic superconductor that can be described
exactly by a Hamiltonian where the only interaction is a nearest-neighbor
attraction. BCS theory can then be applied with arbitrary analytical control,
from which nodeless d-wave or s-wave superconductivity can result.
