We investigate polaron formation when immersing an impurity in a few instances of correlated many-body states. We will review the standard variational treatment, and discuss limitations and extensions of this approach, also by comparison with numerical ED and MPS results. The main finding is that peculiar spectral features allow to distinguish whether the bath features charge density waves, BCS pairing, topological phases, a BKT transition, etc.
Our work is motivated by realizations in cold atom systems as well as recent experiments in few-layer heterostructures, where optical polaron spectroscopy complements transport measurements.