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

Garry Goldstein

TL;DR

The paper tackles the numerical intractability of solving the Hedin equations, which are posed in terms of functional derivatives, by proposing a hierarchy of integral-equation approximations (Hedin I, II, III, …) that promote functional derivatives to independent variables and truncate at a chosen derivative order. In zero-dimensional field theory benchmarks, Hedin I reduces to the GW approximation, while Hedin II and III progressively include higher-order derivative information, leading to substantially richer diagram enumerations and near-exact matching to the full Hedin solution as $n$ increases. The results demonstrate that Hedin II already outperforms state-of-the-art diagrammatic vertex corrections, and Hedin III provides a near-perfect correspondence with the exact zero-D solutions, suggesting strong potential for realistic systems. The work offers a systematic, scalable pathway to improve GW accuracy without resorting to functional-derivative complexity, with promising extensions to DMFT, electron gas, and phonon-enabled problems.

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 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.

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

TL;DR

The paper tackles the numerical intractability of solving the Hedin equations, which are posed in terms of functional derivatives, by proposing a hierarchy of integral-equation approximations (Hedin I, II, III, …) that promote functional derivatives to independent variables and truncate at a chosen derivative order. In zero-dimensional field theory benchmarks, Hedin I reduces to the GW approximation, while Hedin II and III progressively include higher-order derivative information, leading to substantially richer diagram enumerations and near-exact matching to the full Hedin solution as increases. The results demonstrate that Hedin II already outperforms state-of-the-art diagrammatic vertex corrections, and Hedin III provides a near-perfect correspondence with the exact zero-D solutions, suggesting strong potential for realistic systems. The work offers a systematic, scalable pathway to improve GW accuracy without resorting to functional-derivative complexity, with promising extensions to DMFT, electron gas, and phonon-enabled problems.

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 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.

Paper Structure

This paper contains 19 sections, 33 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction
  2. Main Idea and Examples
  3. Hedin approximation I (GW Approximation)
  4. Hedin approximation II
  5. Hedin approximation III
  6. Hedin approximation n
  7. Zero Dimensional Field Theory
  8. Diagrammatic vertex corrections to GW
  9. Self energy for Hedin approximation II
  10. Self energy within Hedin approximation III
  11. Final results
  12. Conclusions and Discussions
  13. Some variations
  14. First variation
  15. Hedin Equations
  16. ...and 4 more sections

Figures (1)

  • Figure 1: Self energy graphs for the GW series, diagrammatic vertex correction, Hedin approximation II, Hedin approximation III and the exact series. We notice that Hedin approximation II is already better (captures the self energy more closely) then the state of the art diagrammatic expansion. Furthermore Hedin III is nearly on top of the exact self energy in zero dimensions and captures nearly all Feynman diagrams.