Systematic improvement of the momentum average approximation for the Green's function of a Holstein polaron

We show how to systematically improve the momentum average (MA) approximation for the Green's function of a Holstein polaron by systematically improving the accuracy of the self-energy diagrams in such a way that they can still all be summed efficiently. This allows us to fix some of the problems of the MA approximation, e.g., we now find the expected polaron+phonon continuum at the correct location, and a momentum-dependent self-energy. The quantitative agreement with numerical data is further improved as expected since the number of exactly satisfied spectral weight sum rules is increased. The corrections are found to be larger in lower dimensional systems.