Nonconvergence of the Feynman-Dyson diagrammatic perturbation expansion of propagators

Abstract

Using a general-order ab initio many-body Green's function method, we numerically illustrate several pathological behaviors of the Feynman-Dyson diagrammatic perturbation expansion of one-particle many-body Green's functions as electron Feynman propagators. (i) The perturbation expansion of the frequency-dependent self-energy is not convergent at the exact self-energy in many frequency domains. (ii) An odd-perturbation-order self-energy has a qualitatively wrong shape and, as a result, many roots of the corresponding Dyson equation are nonphysical in that the poles may be complex or residues can exceed unity or be negative. (iii) A higher even-order self-energy consists of vertical lines at many frequencies, predicting numerous phantom poles with zero residues. (iv) Infinite partial resummations of diagrams by vertex or edge renormalization tend to exacerbate these pathologies. (v) The nonconvergence is caused by the nonanalyticity of the rational-function form of the exact Green's function at many frequencies, where the radius of convergence of its Taylor expansion is zero. This is consistent with the fact that (vi) Pad\'{e} approximants (power-series expansions of a rational function) can largely restore the correct shape and poles of the Green's function. Nevertheless, not only does the nonconvergence render higher-order Feynman-Dyson diagrammatic perturbation theory useless for many lower-lying ionization or higher-lying electron-attachment states, but it also calls into question the validity of its combined use with the ans\"{a}tze requiring the knowledge of all poles and residues. Such ans\"{a}tze include the Galitskii-Migdal identity, the self-consistent Green's function methods, and some models of the algebraic diagrammatic construction.