  A mobile impurity immersed in a non-interacting Fermi sea is dressed by the
gapless particle-hole excitations of the fermionic medium. This conventional
Fermi-polaron setting is well described by the so-called ladder approximation,
which consists in neglecting impurity-hole scattering processes. In this work,
we analyze polaron formation in the context of insulating states of matter,
considering increasing levels of correlation in the medium:~band insulators
originating from external periodic potentials, spontaneously-formed charge
density waves, and a Fermi-Hubbard system undergoing a metal-Mott insulator
transition. The polaron spectral function is shown to exhibit striking
signatures of the underlying fermionic background, such as the single-particle
band gap, particle-hole symmetry and the transition to the Mott state. These
signatures are identified within the framework of the Chevy ansatz, i.e. upon
restricting the Hilbert space to single particle-hole excitations.
Interestingly, we find that the ladder approximation is inaccurate in these
band systems, due to the fact that the particle and hole scattering phase
spaces are comparable. Our results provide a step forward in the understanding
of polaron formation in correlated many-body media, which are relevant to both
cold-atom and semiconductor experiments.
