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Fictitious fluxes in doped antiferromagnets

In a tight binding model of charged spin-1/2 electrons on a square lattice, a fully polarized ferromagnetic spin configuration generates an apparent U(1) flux given by $2π$ times the skyrmion charge density of the ferromagnetic order parameter. We show here that for an antiferromagnet, there are two ``fictitious'' magnetic fields, one staggered and one unstaggered. The staggered topological flux per unit cell can be varied between $-π\leΦ\leπ$ with a negligible change in the value of the effective nearest neighbor coupling constant whereas the magnitude of the unstaggered flux is strongly coupled to the magnitude of the second neighbor effective coupling.