Going beyond the cumulant approximation: Power series correction to the single-particle Green's function in the Holstein system

In the context of a single-electron two orbital Holstein system coupled to dispersionless bosons, we develop a general method to correct the single-particle Green's function using a power series correction (PSC) scheme. We outline the derivations of various flavors of cumulant approximation through the PSC scheme explaining the assumptions and approximations behind them. Finally, we compare the PSC spectral function with cumulant and exact diagonalized spectral functions and elucidate three regimes of this problem - two where the cumulant explains and one where the cumulant fails. We find that the exact and the PSC spectral functions match within spectral broadening across all three regimes.