  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.
