Green's Functions in Many-Body Physics

Core Idea

The single-particle Green's function G(k, omega) = 1/(omega - epsilon_k - Sigma(k, omega)) encodes the propagation of a single electron (or hole) through an interacting many-body system. The self-energy Sigma(k, omega) captures all interaction effects: its real part shifts the quasiparticle energy, and its imaginary part gives the quasiparticle lifetime. The spectral function A(k, omega) = -Im G(k, omega)/pi, which is directly measured by ARPES (angle-resolved photoemission spectroscopy), shows a sharp quasiparticle peak in a Fermi liquid and broad, incoherent features in strongly correlated systems. Feynman diagram techniques provide systematic perturbative approximations for Sigma.

Explainer

Green's functions are the language of many-body quantum physics in condensed matter. While the wavefunction of N interacting electrons is hopelessly complex, the single-particle Green's function G(k, omega) extracts precisely the information relevant to experiments that add or remove one electron: photoemission, tunneling, transport, and optical absorption. It is defined as the Fourier transform of the time-ordered expectation value G(k, t-t') = -i<T c_k(t) c^dagger_k(t')>, where T denotes time ordering and the expectation value is taken in the interacting ground state.

For non-interacting electrons, G_0(k, omega) = 1/(omega - epsilon_k + i delta sgn(epsilon_k - E_F)) has simple poles at the bare band energies. Interactions modify G through the self-energy Sigma(k, omega), via the Dyson equation: G(k, omega) = 1/(omega - epsilon_k - Sigma(k, omega)). The self-energy is the sum of all "proper" (one-particle irreducible) interaction diagrams. Its real part shifts the quasiparticle energy: E*(k) = epsilon_k + Re Sigma(k, E*). Its imaginary part gives the quasiparticle decay rate: Gamma = |Im Sigma(k, E*)|, which translates to a lifetime tau = hbar/(2 Gamma).

The spectral function A(k, omega) = -(1/pi) Im G(k, omega) is the observable quantity — it is directly measured by ARPES. In a Fermi liquid, A(k, omega) near the Fermi surface consists of a sharp Lorentzian peak (the quasiparticle, with weight Z < 1 and width proportional to (omega - E_F)²) sitting on top of a broad incoherent background (weight 1-Z). The quasiparticle residue Z = 1/(1 - partial Re Sigma/partial omega) measures how much of the single-particle character survives the dressing by interactions. In copper, Z ~ 0.8; in heavy fermion compounds, Z ~ 0.001; in a Mott insulator, Z = 0 (no quasiparticle).

Computing Sigma is the central technical challenge. Feynman diagrams provide a systematic perturbative expansion: each diagram represents a specific process (electron-hole pair creation, phonon exchange, repeated scattering) and contributes a specific integral to Sigma. The art is in selecting which diagrams to sum. The GW approximation (Sigma = i G W, where W is the screened Coulomb interaction) captures dynamic screening and gives accurate quasiparticle band structures for semiconductors and simple metals. Dynamical mean-field theory (DMFT) maps the lattice problem onto a self-consistent impurity problem, capturing local correlations and the Mott transition. The Green's function framework thus provides a unified language connecting microscopic many-body theory to experimentally measurable quantities.