Taming singularities of the diagrammatic many-body perturbation theory

TL;DR

This paper introduces a Padé approximation-based method to regularize divergent diagrammatic many-body perturbation series, enabling self-consistent spectral function calculations even with complex poles and negative spectral densities.

Contribution

It presents a novel diagrammatic approach combined with Padé regularization to handle divergences in high-order many-body perturbation theory calculations.

Findings

Successfully applied to a model with a core electron and plasmonic excitation

Determined spectral functions self-consistently up to sixth order with 3111 diagrams

Overcame issues of nonsimple poles and negative spectral densities

Abstract

In a typical scenario the diagrammatic many-body perturbation theory generates asymptotic series. Despite non-convergence, the asymptotic expansions are useful when truncated to a finite number of terms. This is the reason for popularity of leading-order methods such as $GW$ approximation in condensed matter, molecular and atomic physics. Emerging higher-order implementations suffer from the appearance of nonsimple poles in the frequency-dependent Green's functions and negative spectral densities making self-consistent determination of the electronic structure impossible. Here a method based on the Pad\'e approximation for overcomming these difficulties is proposed and applied to the Hamiltonian describing a core electron coupled to a single plasmonic excitation. By solving the model purely diagrammatically, expressing the self-energy in terms of combinatorics of chord diagrams, and regularizing the diverging perturbative expansions using the Pad\'e approximation the spectral function is determined self-consistently using 3111 diagrams up to the sixth order.