Hedin Approximations in Many-Body Physics

• Hedin Approximations are systematic many-body schemes that simplify Hedin’s coupled equations by truncating functional derivatives for iterative solutions.
• They capture increasing sets of Feynman diagrams—from GW to advanced vertex corrections—thereby enhancing accuracy in modeling screening and quasi-particle effects.
• The approach bridges methods like DMFT and parquet theories, ensuring convergence towards the exact solution in both lattice and impurity frameworks.

Hedin approximations denote a systematic hierarchy of self-consistent, many-body schemes for approximating the exact solution to Hedin's equations, which are themselves an exact, closed set of five coupled equations governing the one-particle Green’s function, self-energy, screened interaction, polarization, and vertex function in correlated quantum systems. These approximations provide a controlled, convergent sequence—from GW (the lowest-order case) to higher-order treatments—that systematically incorporate vertex corrections and nontrivial feedback between fermionic and collective bosonic degrees of freedom. The methodology generalizes to electron-phonon, superconducting, spin-dependent, and photon-coupled systems, and has become central to modern computational condensed matter theory (Goldstein, 21 Dec 2025).

1. The Structure and Motivation for Hedin Approximations

The original Hedin equations are written as a set of coupled integral and functional-derivative equations: $$\begin{aligned} \Gamma &= 1 + \frac{\delta \Sigma}{\delta G}\, G\,G\,\Gamma \ P &= -i\,G\,\Gamma\,G \ W &= v + v\,P\,W \ \Sigma &= i\,G\,W\,\Gamma \ G &= g + g\,\Sigma\,G \end{aligned}$$ where $G$ is the fully interacting single-particle Green’s function, $\Sigma$ the self-energy, $W$ the dynamically screened interaction, $P$ the polarization, $\Gamma$ the irreducible three-point (vertex) function, $v$ the bare interaction, and $g$ the Hartree Green’s function. Solving these equations exactly is numerically prohibitive for realistic systems due to the presence of functional derivatives (e.g., $\delta \Sigma/\delta G$).

Hedin approximations—parametrized by a truncation order $n$—systematically convert these functional-differential equations into coupled integral equations for a finite, closed set of objects, enabling practical, convergent iterative solutions. Each level in the sequence, denoted "Hedin I", "Hedin II", "Hedin III", etc., captures additional classes of Feynman diagrams for the self-energy and vertex functions (Goldstein, 21 Dec 2025).

2. The Hedin Approximation Hierarchy and Iterative Schemes

The first three Hedin approximations are as follows:

Order	Main Features	Diagrams Included
I	$\Gamma=1$ (GW)	Bubble (RPA), ring diagrams
II	$\Gamma$ from $\delta\Sigma/\delta G$; add 1st-order vertex corrections	GW + leading vertex corrections
III	Include up to $\delta^2\Sigma/\delta G^2$	Further vertex classes, closer to exact

Hedin I (GW Approximation):

$$\begin{aligned} \Gamma &= 1 \ P &= G G \ W &= v + v P W \ \Sigma &= G W \ G &= g + g \Sigma G \end{aligned}$$

This yields the familiar GW scheme used for quasi-particle band structures and screening in weak to moderately correlated materials (Goldstein, 21 Dec 2025, Held et al., 2011).

Hedin II (First-Derivative Truncation):

Includes the derivatives $\delta \Gamma/\delta G$, resulting in a closed set of 9 coupled integral equations (no free functional derivatives). This scheme captures more diagrams than GW or diagrammatic vertex-corrected GW, and enumerates partial third-order corrections (Goldstein, 21 Dec 2025).

Higher Hedin Approximations:

Each increment includes more functional derivatives as independent functions, rapidly increasing diagrammatic completeness. Hedin III, for instance, achieves near-perfect agreement with the exact solution in zero-dimensional tests and includes diagram classes missed by both GW and common vertex-corrected schemes (Goldstein, 21 Dec 2025).

Convergence:

In zero-dimensional field theoretic tests, the expansion coefficients for the self-energy produced by Hedin II and III rapidly approach the exact series; Hedin III closely matches all diagrams up to order four in the coupling. This empirical fact suggests efficient convergence in more general many-body contexts (Goldstein, 21 Dec 2025).

3. Diagrammatic Content and Physical Meaning

Systematic Hedin approximations classify and sum increasingly complex sets of Feynman diagrams:

• GW (I) sums bubble and ring diagrams (RPA and screened exchange).
• GW plus first-derivative (II) adds ladder-type vertex corrections and select third-order processes.
• Hedin III and higher approximate the full ladder and parquet-type diagrams, key for capturing spin fluctuations, local-moment formation, and nonlocal dynamical screening, especially crucial in strongly correlated or low-dimensional systems (Goldstein, 21 Dec 2025, Krien et al., 2019, Harkov et al., 2021).

The improved expansion captures both single- and multi-boson exchange (including Maki-Thompson and Aslamazov-Larkin diagrams at higher orders), essential for accurate treatment beyond GW, as confirmed by direct diagram counting (Goldstein, 21 Dec 2025).

4. Numerical Algorithms and Practical Performance

Each Hedin approximation can be solved with a closed iterative cycle:

1. Initialize all unknowns (Green's functions, self-energies, vertexes, and their derivatives up to order $n$).
2. Sequentially update $G$, $W$, $\Sigma$, $P$, and $\Gamma$ via the respective integral equations.
3. Iterate derivatives (e.g., $\delta\Gamma/\delta G$) according to the truncated chain rule expressions.
4. Iterate to self-consistency using mixing or DIIS acceleration.

Numerical tests in zero-dimensional theories and impurity models demonstrate rapid convergence even in moderately correlated regimes, with Hedin II/III outperforming state-of-the-art external diagrammatic correction schemes in diagram coverage and accuracy (Goldstein, 21 Dec 2025, Krien et al., 2019).

5. Comparison to Other Many-Body Approaches

Hedin approximations provide an alternative to conventional approaches:

• Parquet Equations: The full parquet scheme is exact but scales as $O(N^3)$ in Matsubara frequency variables, vastly increasing computational cost. SBE-based (single-boson-exchange) Hedin approximations avoid Bethe-Salpeter equations and reduce scaling to $O(N^2)$, while preserving vital crossing symmetries (Krien et al., 2019).
• Ladder Theories: Simpler ladder or T-matrix approximations neglect multi-channel feedback. Hedin approximations systematically include these, bridging the gap between ladders and full parquet solutions.
• GW+DMFT and D$\Gamma$A: Hedin approximations interact constructively with DMFT-based local vertex modeling, as in dynamical vertex approximation (D$\Gamma$A). The SBE variant parametrizes the full vertex hierarchy in terms of local (impurity) three-leg vertices, inheriting strong-correlation physics at reduced computational expense (Harkov et al., 2021, Krien et al., 2019).

6. Physical Implications and Limitations

The systematic approach guarantees that with increasing truncation order, the approximate solution converges to the exact Hedin solution, as neglected higher-order functional derivatives become progressively less relevant. In practice:

• Hedin II captures more physical diagrams for the self-energy than external diagrammatic vertex corrections or simple ladder resummations.
• Hedin III achieves near-exact self-energy expansions up to high order in zero-dimensional benchmarks (Goldstein, 21 Dec 2025).
• In realistic lattice or impurity settings, the scheme captures local and nonlocal fluctuations important for magnetism, Mott transitions, unconventional superconductivity, and ultrafast dynamics.

Current limitations include increased algebraic complexity at higher orders and the requirement for appropriate boundary conditions and accurate local (impurity) vertex inputs when working with lattice models or in DMFT extensions (Goldstein, 21 Dec 2025, Krien et al., 2019).

7. Extensions and Outlook

Hedin approximations have been generalized to encompass:

• Nonequilibrium states, via closed time-path (Keldysh) extensions, and to coupled electron-phonon systems, with appropriate reshaping of self-energies and vertices to preserve conservation laws (Stefanucci et al., 2023).
• Systems exhibiting superconductivity (using generalized Nambu-Gor’kov and anomalous Green’s functions) (Linscheid et al., 2015, Lane, 8 Jun 2025).
• Strongly coupled cavity quantum electrodynamics, where the photonic degrees of freedom are embedded in the Hedin hierarchy (Trevisanutto et al., 2015).
• Spin-dependent and relativistic effects, with the vertex and $W$ extended to tensor structures in spin-charge space (Lane, 8 Jun 2025).

The convergent and integral-equation nature of Hedin approximations makes them particularly suitable for future developments in high-accuracy ab initio many-body calculations for real materials, systems under external drive, and in combined approaches that interface with DMFT, D$\Gamma$A, and parquet solvers (Goldstein, 21 Dec 2025, Held et al., 2011, Krien et al., 2019).