| Green's function (many-body theory) | |
|---|---|
| Name | Green's function (many-body theory) |
| Other names | Many-body Green's function, propagator |
| Field | Quantum mechanics; Condensed matter physics |
| Introduced | 1950s |
| Notable works | LSZ formalism, Feynman diagram |
| Practitioners | Bohm, Schwinger, Feynman, Landau |
Green's function (many-body theory)
A many-body Green's function is a correlation function or propagator used in quantum mechanics and condensed matter physics to describe the propagation and interactions of excitations in a quantum many-particle system. It encodes spectral, dynamical and thermodynamic information and provides a unifying framework for calculating observable quantities such as excitation spectra, response functions and transport coefficients. Many-body Green's functions underpin computational methods in electronic structure and are central to theoretical approaches developed by Schwinger, Feynman, and later many-body theorists.
Many-body Green's functions generalize the concept of single-particle propagators to interacting systems of fermions or bosons, allowing correlation effects beyond independent-particle models like Hartree–Fock. They connect microscopic Hamiltonians (for example the Hubbard model or realistic electronic Hamiltonians used in density functional theory studies) to experimentally measurable quantities, such as photoemission spectroscopy spectra and linear response coefficients like optical conductivity. Green's functions provide a compact description of causality, statistics (Fermi–Dirac or Bose–Einstein), and conservation laws, and serve as the starting point for systematic approximations and renormalization techniques used in many-body perturbation theory.
For fermionic field operators ψ and ψ† in the Heisenberg picture, the single-particle time-ordered Green's function is defined as G(1,2) = −i⟨T{ψ(1)ψ†(2)}⟩, where T is the time-ordering operator and numbers denote space, spin and time coordinates. At finite temperature one often uses the Matsubara formalism on the imaginary-time axis (τ) introduced by Matsubara, leading to discrete Matsubara frequencies. Green's functions can be classified as time-ordered, retarded, advanced, greater and lesser functions, each of which is suited to particular physical contexts such as equilibrium, non-equilibrium Keldysh dynamics, or linear response. The first occurrence of key operators and formal constructs connects to foundational works by Schwinger and Kadanoff & Baym.
Green's functions satisfy equations of motion derived from the Heisenberg equation, leading to integro-differential relations. Central is the Dyson equation, G = G0 + G0 Σ G, which relates the interacting Green's function G to the noninteracting propagator G0 and the self-energy Σ that encapsulates exchange and correlation effects. The self-energy is generally nonlocal in space and time and incorporates processes such as quasiparticle renormalization and lifetime broadening first emphasized in Landau theory and later formalized within diagrammatic expansions. The Dyson equation is the basis for self-consistent schemes and for connecting to observable poles and branch cuts of G.
Perturbation theory for Green's functions is commonly organized using Feynman diagram techniques developed for many-body contexts. Wick's theorem reduces time-ordered products to sums of contractions, producing diagrammatic series for Σ and higher-order correlation functions. Diagram classes include Hartree and Fock diagrams, polarization bubbles, vertex corrections, and ladder series relevant for screening and pairing. Diagrammatic Monte Carlo and resummation approaches address convergence, while renormalization-group methods treat scale-dependent fluctuations. Important contributors include Migdal and Eliashberg for electron–phonon interactions and Hedin who introduced the GW set of coupled equations.
The spectral function A(k,ω) = −(1/π) Im G^R(k,ω) encodes the density of excitations and is directly linked to spectra measured by techniques such as ARPES. Analytic continuation from imaginary to real frequencies uses complex analysis and causality: retarded and advanced Green's functions are analytic in complementary half-planes. The Lehmann representation expresses G in terms of exact many-body eigenstates and energies, rendering explicit the relation between poles/weights and quasiparticle excitations. Sum rules and moment expansions constrain spectral weight distributions and are frequently used to validate approximations.
Practical calculations require controlled approximations for the self-energy. The Hartree–Fock approximation represents the leading mean-field exchange, while the GW approximation (Hedin) computes Σ ≈ i GW with a dynamically screened interaction W and has become a standard for quasiparticle energies in materials science. The DMFT maps lattice models to an impurity problem to capture local dynamical correlations, often combined with density functional theory in DFT+DMFT workflows. Other methods include second-order perturbation theory, parquet approximations, and self-consistent schemes such as GW+DMFT. Vertex corrections and conserving approximations (Baym–Kadanoff) ensure conservation laws are respected.
Many-body Green's functions are applied across condensed matter and quantum chemistry: computing quasiparticle band structures, spectral functions, optical absorption, and transport coefficients such as electrical and thermal conductivity. They are used to model correlated materials (e.g., transition-metal oxides and heavy-fermion compounds), superconductivity through anomalous Green's functions and the Eliashberg theory, and non-equilibrium dynamics in ultrafast pump–probe experiments via the Keldysh formalism. Implementations appear in codes and projects such as Quantum ESPRESSO, VASP, and community efforts in electronic-structure theory, linking ab initio Hamiltonians to measurable response functions and guiding interpretation of experiments like X-ray photoelectron spectroscopy and inelastic neutron scattering.
Category:Quantum mechanics Category:Condensed matter physics