Trapping of three-dimensional Holstein polarons by various impurities

We study the bound states of a three-dimensional Holstein polaron near various kinds of single impurities, using the momentum-average approximation. We show that the electron-phonon coupling renormalizes the impurity potential into a strongly retarded effective potential, which describes essential physics ignored by ``instantaneous'' approximations. The accuracy of our method is gauged by comparison with results from diagrammatic Monte Carlo for the case of an impurity that modifies the on-site energy of the electron. We also discuss impurities that modify the local strength of the electron-phonon coupling as well as isotope substitutions that change both the electron-phonon coupling and the phonon frequency. For the latter, we recover a threshold value of the electron-phonon coupling below which, no matter how strong the impurity is, polaron can not be trapped.