Effective interactions between local hopping modulations on the square lattices

We have computed the Ruderman-Kittel-Kasuya-Yosida (RKKY) interaction between Ising spins variables living on the links of the square lattice. The interaction is mediated by itinerant fermions, which couple to the Ising spins in the form of a hopping modulation. We consider the case of two different external magnetic fields: B = 0 (zero-flux), that produces a conventional Fermi surface, and B ≠ 0 (π-flux), that gives Dirac points. The RKKY interaction between Ising spins on the links is contrasted with the known result for spins placed at each lattice vertex. For the zero-flux square lattice, the strong nesting property of the Fermi surface at half-filling determines the ordering vector Q_o = (π, π). We discuss the different magnetic orderings that arise in the π-flux, Dirac-fermion case. This is relevant to recently introduced models in which strong interactions between the link variables can drive a quantum transition with topological character.