| Green's functions (many-body theory) | |
|---|---|
| Name | Green's functions (many-body theory) |
| Field | Theoretical physics |
| Introduced | 1950s |
| Notable users | Lev Landau, John Bardeen, Richard Feynman, Julian Schwinger |
Green's functions (many-body theory)
Green's functions (many-body theory) are correlation functions used to describe propagation and response in interacting quantum systems. They generalize classical Green's functions to the quantum and many-body problem contexts, providing access to spectra, excitation lifetimes, and response functions. Green's functions underpin computational and conceptual frameworks across condensed matter physics, nuclear physics, and quantum chemistry.
In many-body quantum physics a Green's function encodes expectation values of time-ordered products of field operators and thereby captures how single-particle or collective excitations propagate in an interacting background. The method links microscopic Hamiltonians (e.g. the Hubbard model or electron–phonon interaction) to experimentally observable quantities such as angle-resolved photoemission spectroscopy (ARPES) spectra, optical conductivities, and scattering cross sections. Historically, the machinery was systematized by work of Julian Schwinger, Richard Feynman, and others building on techniques from Landau's Fermi liquid theory. Green's functions are valued for their ability to maintain consistency with conservation laws and to incorporate perturbative and nonperturbative corrections in a unified formalism.
Standard two-point Green's functions include the time-ordered (or causal) propagator G_T, the retarded propagator G_R, and the advanced propagator G_A. For fermions these are defined from field operators in the Heisenberg picture as expectation values over a many-body state or thermal ensemble (grand canonical density matrix). The time-ordered Green's function relates directly to perturbation theory in the interaction picture used by Richard Feynman's diagrammatic approach, while the retarded Green's function connects to linear response theory as formulated by Kubo and used in transport calculations. Analytic continuation between real-time and imaginary-time (Matsubara) Green's functions is central for finite-temperature calculations pioneered by Alexander Abrikosov and Gordon Baym.
The formal derivation uses second quantization with creation and annihilation operators acting in a Fock space built from single-particle states (often Bloch states in crystals or harmonic oscillator states in nuclei). Field operators ψ†(x,t), ψ(x,t) create and annihilate particles at spacetime points; Green's functions are correlation functions like ⟨T ψ(x,t) ψ†(x',t')⟩. Many concrete implementations reference model Hamiltonians such as the Anderson impurity model, BCS Hamiltonian for superconductivity, or realistic electronic structure Hamiltonians used in quantum chemistry methods (e.g., coupled-cluster or configuration interaction where Green's functions provide excited-state information).
Perturbative evaluation is organized via the diagrammatic expansion developed by Richard Feynman and extended by many-body practitioners. Diagrams represent propagators and interaction vertices; combinatorial rules produce algebraic expressions for Green's functions in terms of integrals over internal frequencies and momenta. Important classes of diagrams include bubble diagrams (screening), ladder diagrams (vertex corrections and pairing), and ring diagrams (collective modes). Diagrammatic techniques tie to computational methods such as Quantum Monte Carlo when stochastic sampling of diagrammatic series is employed, and to resummation approaches like the random phase approximation (RPA) and parquet equations.
The Dyson equation relates the full interacting Green's function G to a noninteracting reference G0 via the self-energy Σ: G = G0 + G0 Σ G. The self-energy contains all irreducible interaction effects and determines renormalization of quasiparticle energies and lifetimes, as well as spectral weight transfer. Approximations for Σ range from perturbative GW approximation (Hedin) to conserving approximations of Baym and Kadanoff and nonperturbative methods like dynamical mean-field theory (DMFT) which maps lattice problems to impurity models solved self-consistently (notably used by groups at Argonne National Laboratory and ETH Zurich).
The imaginary part of the retarded Green's function yields the spectral function A(k,ω), directly measurable in ARPES and neutron scattering. Spectral functions obey analytic properties following causality and Kramers–Kronig relations; moments of A(k,ω) satisfy sum rules determined by commutation relations and particle number conservation. Lehmann representations express Green's functions in terms of exact many-body eigenstates and energies, clarifying poles (quasiparticles) versus branch cuts (continuum excitations). These analytic structures guide numerical analytic continuation from Matsubara to real frequencies, a central challenge addressed by techniques like maximum entropy and Padé approximants.
In condensed matter physics Green's functions diagnose Fermi surface properties, superconductivity, magnetism, and collective modes in materials such as high-temperature superconductors and heavy-fermion compounds studied at institutions like Max Planck Institute for Chemical Physics of Solids. In nuclear physics they describe propagations of nucleons and collective excitations, underpinning theories of nuclear structure and reactions in laboratories like CERN and Oak Ridge National Laboratory. In quantum chemistry Green's function approaches (e.g., GW, equation-of-motion techniques) complement wavefunction methods to compute ionization potentials, electron affinities, and excitation spectra for molecules and materials, used in software developed by research groups at Brookhaven National Laboratory and university centers. Cross-disciplinary collaborations and national research programs emphasize Green's functions as a stable, unifying tool to predict and interpret experiments while supporting technological and educational missions.
Category:Quantum mechanics Category:Many-body physics