Green's Functions

Green’s functions provide a compact language for asking how an added electron or hole propagates through an interacting system. Rather than describing a single wavefunction, the one-particle Green’s function organizes excitation energies, spectral weights, lifetimes, densities, and thermodynamic information in one frequency-dependent object.

The central objects

For a non-interacting reference with one-particle Hamiltonian $h_0$, the frequency-domain propagator is

$$ G_0(z)=\left[(z+\mu)S-h_0\right]^{-1}, $$

where $S$ is the overlap matrix and $\mu$ is the chemical potential. Interactions dress this propagator through Dyson’s equation,

$$ G^{-1}(z)=G_0^{-1}(z)-\Sigma(z). $$

The self-energy $\Sigma$ is the effective, energy-dependent potential that contains exchange and correlation. Choosing an approximation to $\Sigma$, then solving Dyson’s equation consistently, defines a Green’s-function method.

Approximations and extensions in this notebook

A self-consistent calculation

1. Build a reference Green’s function $G_0$ and choose $\mu$.
2. Evaluate the self-energy $\Sigma[G]$ for the selected approximation.
3. Solve Dyson’s equation to obtain an updated $G$.
4. Update the density and chemical potential, then repeat until all target quantities converge.
5. Continue $G$ or $\Sigma$ from the imaginary axis when real-frequency spectra are required.

The same loop can behave very differently under GF2, GW, and vertex-corrected GW because their diagrammatic content, screening, and failure modes differ. The notes below focus on those distinctions.