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I will discuss the Momentum Average approximation -- a simple, highly
efficient yet accurate analytical approximation for the Green's function of
a Holstein polaron. One way of explaining it is that it corresponds to
summing all of the self-energy diagrams, but with each self-energy diagram
averaged over the momenta of its free propagators. The result becomes exact
for both zero bandwidth and for zero electron-phonon coupling, and is
accurate everywhere in the parameter space. The resulting Green's function
satisfies exactly the first six spectral weight sum rules, and all higher
sum rules are satisfied with great accuracy. Comparison with existing
numerical data also confirms this accuracy. A systematic way to improve the
accuracy of the approximation will then be introduced. Finally, I will
briefly discuss generalizations to other models that we have successfully
carried out to date.
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