Sum rules satisfied by Feynman-approximation solutions to the polaron problem

Abstract

A free-energy theorem for polarons is derived as an extension to arbitrary lattice temperature of the ground-state theorem derived by Lemmens, deSitter, and and Devreese for zero lattice temperature. These theorems relate the exact free energy of the polaron to the exact absorption spectrum integrated over frequency. With approximate solutions in general, and with approximate solutions to many-body problems in particular, it is of interest to know how closely such sum rules are satisfied. We prove that the free-energy theorem is valid for any Feynman-approximation solution to the polaron problem. Using this analytic result, we are able to determine how sensitively satisfying the free-energy theorem depends upon the details of the absorption spectrum. Considering examples of two quite different spectra at zero temperature, we conclude that the ground-state theorem is of little value in discriminating between different calculated absorption spectra. A more sensitive alternative is discussed which makes use of the fluctuation-dissipation theorem for comparing the calculated absorption with the fluctuation energy at each frequency. Since this alternative method requires equality between two functions for each value of their argument (frequency), rather than merely between two numbers as with sum rules, the method easily distinguishes between spectra, each of which may satisfy the ground-state rule very closely. This method is closely related to a self-consistent procedure developed previously. The latter procedure removed the ambiguity of which approximate influence functional to use in path-integral treatments of polaronlike problems.