Application of the generalized 3D Jordan-Wigner transformation to the bilayer Heisenberg antiferromagnet

We extend the definition of the Jordan-Wigner transformation to three dimensions using the generalization of ideas that were used in the two-dimensional case by one of the present authors. Under this transformation, the 3D XY Hamiltonian is transformed into a system of spinless fermions coupled to a gauge field with only two components. We calculate the flux per plaquette for the 3 elementary perpendicular plaquettes of a cubic lattice, and find that it is nonzero for only two of the plaquettes. We provide a simple interpretation for the average phase-per-plaquette being $\pi$ on the plaquettes where it is nonzero. Then we apply these findings to the investigation of the Heisenberg bilayer antiferromagnet.