Perturbational study of the lifetime of a Holstein polaron in the presence of weak disorder

Using the momentum average (MA) approximation, we find an analytical expression for the disorder-averaged Green's function of a Holstein polaron in a three-dimensional simple cubic lattice with random on-site energies. The on-site disorder is assumed to be weak compared to the kinetic energy of the polaron, and is treated perturbationally. Within this scheme, the states at the bottom of the polaron band are found to have an infinite lifetime, signaling a failure of perturbation theory at these energies. The higher-energy polaron states have a finite lifetime. We study this lifetime and the disorder-induced energy shift of these eigenstates for various strengths of disorder and electron-phonon coupling. We compare our findings to the predictions of Fermi's golden rule and the average T-matrix method, and find a significant quantitative discrepancy at strong electron-phonon coupling, where the polaron lifetime is much shorter than Fermi's golden rule prediction. We attribute this to the renormalization of the on-site potential by the electron-phonon coupling.