Transformation from Eliashberg theory to polarons.

We present the results of numerical and analytical (diagrammatic) studies of the emergence of polarons in a system of spinless fermions interacting with Einstein phonons. We consider electronic densities n between n=0 and n=1 and for each n vary the dimensionless coupling l. We show that for all n, a polaron emerges as a bound state at l of order one (largest for n =1/2) , and gives rise to a jump in the chemical potential and strong reconstruction of low-energy states. We compare the coupling at which polarons emerge with the end point of the applicability of Eliashberg theory and argue that they match near n=1/2, but not at smaller or larger n.