  We investigate the Fermi polaron problem in a spin-1/2 Fermi gas in an
optical lattice for the limit of both strong repulsive contact interactions and
one dimension. In this limit, a polaronic-like behaviour is not expected, and
the physics is that of a magnon or impurity. While the charge degrees of
freedom of the system are frozen, the resulting tight-binding Hamiltonian for
the impurity's spin exhibits an intriguing structure that strongly depends on
the filling factor of the lattice potential. This filling dependency also
transfers to the nature of the interactions for the case of two magnons and the
important spin balanced case. At low filling, and up until near unit filling,
the single impurity Hamiltonian faithfully reproduces a single-band,
quasi-homogeneous tight-binding problem. As the filling is increased and the
second band of the single particle spectrum of the periodic potential is
progressively filled, the impurity Hamiltonian, at low energies, describes a
single particle trapped in a multi-well potential. Interestingly, once the
first two bands are fully filled, the impurity Hamiltonian is a near-perfect
realisation of the Su-Schrieffer-Heeger model. Our studies, which go well
beyond the single-band approximation, that is, the Hubbard model, pave the way
for the realisation of interacting one-dimensional models of condensed matter
physics.
