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. 
B. Bock (UNBC, Prince George) and M. Azzouz (UNBC, Prince George, and   Max-Planck Ins., Dresden)