  The 2D quantum phase transition that occurs in a square lattice of $p+ip$
superconductors is used to show how four-body interactions in $d=2$ reproduce
nonperturbative effects familiar from the study of two-body interactions in
$d=1$. This model can be analyzed using an exact lattice self-duality of the
associated 3D classical model; this duality is the 3D generalization of the
Kramers-Wannier duality of the 2D Ising model, and there are similar exact
dualities for certain many-spin interactions in dimensions $d \geq 3$. We also
discuss the excitation spectrum in the ordered and disordered phases, and the
relationship between our model and previously studied metallic behavior of
boson models with four-boson interactions.
