Vertex symmetry and the asymptotic frequency dependence of the self-energy

Abstract

We show that the exact 1/${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{n}}$ dependence of the self-energy for the Hubbard model for large ${\mathrm{\ensuremath{\varepsilon}}}_{\mathrm{n}}$ is obtained entirely from the second-order skeleton diagram evaluated with the exact propagator. Explicit calculation demonstrates that contributions from higher-order skeleton diagrams cancel exactly as a consequence of the crossing symmetry of the renormalized four-point vertex function. From an analysis of the skeleton diagram expansion and direct numerical calculation, we show that an imperfect cancellation of contributions from higher-order skeleton diagrams appears at third and fourth order in U for the shielded potential approximation and the fluctuation exchange approximation, respectively. By itself, this indication of crossing symmetry violation of the vertex function says little about the accuracy of the resulting self-energy and propagator, particularly at low frequency.