Many-Body Correlators, Response, and Quasiparticles

Many-body correlators connect particle addition, collective response, and measurable spectra. Their poles and continua reveal different excitations, while conservation laws constrain how self-energies and response vertices may be approximated.

Enter this chapter

Begin with Quantum-Matter Correlators and Observable Conventions to match the measured observable to the appropriate correlator.

Helpful background. Quantum-Matter Correlators and Observable Conventions and Sources, Linear Response, and Kubo Formulae develop the definitions used by the chapter’s opening pages.

Topics

1. Quantum-Matter Correlators and Observable Conventions — Fixes signs, ordering, thermal factors, spectral normalization, and continuation.
2. Lehmann Representations and Spectral Functions in Matter — Expresses addition, removal, and density spectra in exact many-body eigenstates.
3. Dyson Equations and Self-Energies — Defines the exact self-energy and its pole, width, and incoherent contributions.
4. Diagrammatics at Finite Density — Separates vacuum renormalization from medium occupation factors and particle–hole kinematics.
5. Quasiparticle Poles, Residues, and Lifetimes — Gives the scale hierarchy required before calling a spectral feature a particle.
6. Spectral Moments and Many-Body Sum Rules — Derives moment constraints from equal-time commutators and the $f$-sum rule.
7. Polarization, the Lindhard Function, and Particle–Hole Continua — Derives the free Fermi-gas density response and its noncommuting limits.
8. Coulomb Screening, Dielectric Response, and RPA Validity — Resums polarization bubbles and states where that resummation is controlled.
9. Plasmons and Collective Charge Modes — Derives two- and three-dimensional dispersions and the damping boundary.
10. Current Vertices and Ward-Consistent Response — Ties self-energy and response vertices to the continuity equation.
11. Irreducible Vertices and Bethe–Salpeter Equations — Defines channel reducibility and susceptibility eigenvalue problems.
12. Hedin’s Equations, Screened Interactions, and the GW Approximation — Derives the coupled hierarchy and identifies the vertex truncation behind each GW scheme.
13. Luttinger–Ward Functionals and Φ-Derivable Self-Energies — Constructs stationary functionals and conserving self-consistency.
14. Baym–Kadanoff Conservation Laws and Approximation Validity — Separates conservation, thermodynamic consistency, Ward closure, crossing, positivity, and numerical accuracy.

How the ideas fit together

A sharp one-particle pole describes adding an excitation; a density-response pole describes collective motion. Screening and collective modes therefore require vertices as well as dressed propagators. Deriving a response consistently from a self-energy functional links these two levels and preserves the associated conservation laws Baym 1962, pp. 1393–1397.

Conservation is still only one test. A conserving calculation can have inaccurate spectra, inadequate crossing relations, or an uncontrolled truncation. Spectral moments, widths, Ward identities, and comparisons across controlled regimes answer complementary questions.

The comparison below separates the assumptions and tests needed for these different conclusions.

Comparison of claims and tests

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Tests and limits of claims made with many-body Green functions.

Claim	Object and convention	Necessary test	What the test establishes	What it does not establish
Spectral weight is normalized	Canonical one-particle $A=-2\operatorname{Im}G^R$	Zeroth moment from the equal-time anticommutator	Operator normalization and total one-particle weight	Correct placement of the weight
A peak is a quasiparticle	Retarded pole on the physical continuation	Small width, finite residue, stable pole, separated scales	Long-lived one-particle-like excitation	Transport lifetime or Landau-liquid behavior
A density approximation is consistent	$\chi^R_{nn}$ with contact terms	Continuity Ward identity and $f$-sum rule	Conservation and normalization	Quantitative accuracy at all $q,\omega$
RPA screening is controlled	$\varepsilon=1-v\Pi_0$ with neutral background	High-density or large-degeneracy counting plus vertex estimate	Parametric reason bubble chains dominate	Universal reliability at strong coupling
A loss peak is a plasmon	Pole of $\varepsilon^{-1}$ on the retarded sheet	Complex pole, residue, and continuum separation	Collective charge mode with a damping rate	A perfectly sharp excitation
A Bethe–Salpeter eigenvalue is large	Named irreducible channel and kernel	Grid, basis, crossing, and finite-size checks	Enhanced susceptibility in that approximation	Established broken symmetry
A scheme is Φ-derivable	$\Sigma=\delta\Phi/\delta G$ solved self-consistently	Functional symmetry and consistent response vertex	Named macroscopic conservation laws	Crossing, positivity, uniqueness, or accuracy
GW gives a useful spectrum	Declared $G_0W_0$, eigenvalue-only, quasiparticle self-consistent, or fully self-consistent scheme	Starting point, self-consistency, vertex, sum-rule, and benchmark tests	Performance in the tested regime	A general controlled expansion

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Where to go next

Continue to Resonant Interactions and Universal Quantum Gases to match short-range interactions to scattering observables.

References

• Baym, Gordon. “Self-Consistent Approximations in Many-Body Systems.” Physical Review 127 (1962): 1391–1401. DOI.