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We show how to systematically improve the Momentum Average (MA)approximationfor the Green's function of a Holstein polaron, bysystematically improving theaccuracy of the self-energy diagrams in such a way that they can still all besummed efficiently. This allows us to fix some of the problems of the MAapproximation, e.g. we now find the expected polaron+phonon continuum at thecorrect location, and a momentum-dependent self-energy. The quantitativeagreement with numerical data is further improved, as expected since the numberof exactly satisfied spectral weight sum rules is increased. The correctionsare found to be larger in lower dimensional systems.

Systematic improvement of the Momentum Average approximation for the Green's function of a Holstein polaron