A Systematic Convergent Sequence of Approximations (of Integral Equation Form) to the Solutions of the Hedin Equations

TL;DR

This paper introduces a sequence of integral equation-based approximations to the Hedin equations, systematically improving convergence and diagram enumeration, offering a numerically tractable alternative to traditional functional derivative approaches.

Contribution

It presents a systematic set of integral equations (Hedin approximations I-IV) that converge to the exact Hedin solutions, improving upon the GW approximation and capturing more Feynman diagrams.

Findings

Hedin approximation I is equivalent to the GW approximation.

Higher Hedin approximations capture more Feynman diagrams.

Hedin approximation III closely matches exact solutions in zero-dimensional theory.

Abstract

In many ways the solution to the Hedin equations represents an exact solution to the many body problem. However, for most systems of practical interest, the solution to the Hedin equations is rendered nearly numerically intractable because the Hedin equations are of functional derivative form. Integral equations, on the other hand, are much more numerically tractable, than functional derivative equations, as they can often be solved iteratively. In this work we present a systematic set of integral equations (with no functional derivatives) - Hedin approximations I, II, III, IV etc. - whose solutions converge to the solutions of the exact Hedin equations. The Hedin approximations are well suited to iterative numerical solutions (which we also describe). Furthermore Hedin approximation I is just the GW approximation (as such this work may be viewed as a systematic improvement of the GW approximation). We present a systematic study of the Hedin equations for zero dimensional field theory (which, in particular, is a method to enumerate Feynman diagrams for field theories in arbitrary dimensions) and show better and better convergence to the solutions of the Hedin equations for higher and higher Hedin approximations, with Hedin approximations I, II and III being explicitly studied. We, in particular, show that the higher Hedin approximations capture more and and more Feynman diagrams for the self energy. We also show that already Hedin approximation II captures more diagrams than the state of the art diagrammatic vertex corrections approach. Furthermore Hedin approximation III is a near perfect match to the exact solutions of the Hedin equations, at least in the zero dimensional case, and enumerates a large number of Feynman diagrams.