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
π 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.
Abstract Utilizing determinant quantum Monte Carlo algorithm, the evolution of the d-wave pairing in the Hubbard model on the square lattice tuned by the periodically modulated hopping amplitudes is studied. The hopping amplitudes are homogeneous in the x ˆ -direction, while in the y ˆ -direction the hopping amplitudes are modulated with period P, where t y = t + d t , t y ′ = t − ( P − 1 ) d t , and the modulation period P equals 2, 3 and 4 lattice spacings. The latter two modulation periods are motivated by the observation of period-3 and period-4 stripe order in cuprate superconductors. For all the periods P, we find that the modulated hopping inhomogeneity enhances the d-wave pairing and an optimal inhomogeneity exists.