  A combined analytical and numerical study is performed of the mapping between
strongly interacting fermions and weakly interacting spins, in the framework of
the Hubbard, t-J and Heisenberg models. While for spatially homogeneous models
in the thermodynamic limit the mapping is thoroughly understood, we here focus
on aspects that become relevant in spatially inhomogeneous situations, such as
the effect of boundaries, impurities, superlattices and interfaces. We consider
parameter regimes that are relevant for traditional applications of these
models, such as electrons in cuprates and manganites, and for more recent
applications to atoms in optical lattices. The rate of the mapping as a
function of the interaction strength is determined from the Bethe-Ansatz for
infinite systems and from numerical diagonalization for finite systems. We show
analytically that if translational symmetry is broken through the presence of
impurities, the mapping persists and is, in a certain sense, as local as
possible, provided the spin-spin interaction between two sites of the
Heisenberg model is calculated from the harmonic mean of the onsite Coulomb
interaction on adjacent sites of the Hubbard model. Numerical calculations
corroborate these findings also in interfaces and superlattices, where
analytical calculations are more complicated.
