Effective interactions between local hopping modulations on the square lattice

We address the problem of free fermions interacting with frozen gauge fields. In particular, we consider a tight-binding model of fermions on the square lattice in which (i) flux 0 or $\pi$ is threaded through each plaquette and (ii) each nearest-neighbor link is decorated with an Ising degree of freedom that describes the local modulation of the hopping amplitude. Following the standard Ruderman--Kittel--Kasuya--Yosida (RKKY) approach, we compute an effective spin model in the coupling strength order by order. Unlike the original RRKY result for site-centered SU(2) spins in which the leading contribution is an effective exchange term at the second-order, perturbation theory in link-centered Z$_2$ case produces a first-order term that favors a collective ferromagnetic moment. If, by some means, an antiferromagnetic configuration can be stabilized, the energetics of ground state is controlled by an effective Ising interaction acting pairwise at the long range across the system.