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

Within the path-integral formalism, we derive exact expressions for correla tion functions measuring the lattice charge induced by an electron and asso ciated polarization in the Frohlich polaron problem. We prove that a sum ru le for the total induced charge, already obtained with approximate approach es, is indeed exact. As a consequence the total induced charge is shown rig orously to be temperature independent. In addition, we perform path integra l Monte Carlo calculations of the correlation functions and compare them to variational results based on the Feynman method. As the temperature increa ses the polaron radius decreases. On the other hand, at high temperatures t he electron motion is not hindered by the lattice. These apparently contrad ictory results are discussed.