Induced charge in a Fröhlich polaron: Sum rule and spatial extent

Within the path-integral formalism, we derive exact expressions for correlation functions measuring the lattice charge induced by an electron and associated polarization in the Fr\"ohlich polaron problem. We prove that a sum rule for the total induced charge, already obtained with approximate approaches, is indeed exact. As a consequence the total induced charge is shown rigorously to be temperature independent. In addition, we perform path integral Monte Carlo calculations of the correlation functions and compare them to variational results based on the Feynman method. As the temperature increases the polaron radius decreases. On the other hand, at high temperatures the electron motion is not hindered by the lattice. These apparently contradictory results are discussed.