The retarded Green's function G^R(k, t) = -iθ(t)<{c_k(t), c†_k(0)}> describes the propagation of an added electron. What physical question does it answer?
AIt tells you the energy of the electron in the crystal
BIf you add an electron with momentum k at time 0 to the interacting ground state, G^R(k, t) gives the amplitude that the system still has that electron with momentum k at time t. In frequency space, the poles of G(k, ω) give the quasiparticle energies and lifetimes — the complete excitation spectrum of the interacting system as seen by adding or removing one electron
CIt describes the scattering of two electrons
DIt gives the pair correlation function between electrons
The Green's function is the fundamental propagator of the many-body system. It tells you everything about single-particle-like excitations: their energies (pole positions), lifetimes (pole widths), and spectral weight (residues). For a non-interacting system, G₀(k,ω) = 1/(ω - ε_k + iδ) has poles at the bare band energies with infinite lifetime. Interactions move the poles (renormalize energies), broaden them (finite lifetime), and redistribute spectral weight from the quasiparticle peak to an incoherent background.
Question 2 Multiple Choice
The self-energy Σ(k, ω) is the central object in many-body perturbation theory. If you know Σ exactly, what do you know?
AOnly the electron-phonon coupling strength
BEverything about single-particle excitations: the full Green's function G = 1/(ω - ε_k - Σ), from which you extract the quasiparticle dispersion E*(k) = ε_k + Re Σ(k, E*), the quasiparticle lifetime τ = ħ/|2 Im Σ|, the quasiparticle residue Z = (1 - ∂Re Σ/∂ω)^{-1}, and the spectral function A(k,ω). The self-energy is the complete encoding of all many-body effects on single-particle propagation
CThe total energy of the system
DThe self-energy only gives the effective mass
The Dyson equation G = G₀ + G₀ΣG means that Σ is the 'correction' to free propagation caused by interactions. Knowing Σ exactly gives the exact single-particle Green's function and thus the exact spectral function, quasiparticle properties, and single-particle density of states. In a Fermi liquid, Im Σ ~ (ω - E_F)² near E_F (giving long-lived quasiparticles), and the quasiparticle residue Z < 1 measures the fraction of spectral weight in the coherent peak versus the incoherent background.
Question 3 True / False
ARPES (angle-resolved photoemission spectroscopy) measures the spectral function A(k, ω) = -(1/π)Im G(k, ω). In a Fermi liquid, A(k, ω) shows a sharp peak (quasiparticle) on a broad background (incoherent spectral weight).
TTrue
FFalse
Answer: True
This is the direct experimental test of many-body theory. In a non-interacting system, A(k,ω) is a delta function at ω = ε_k. In a Fermi liquid, interactions broaden the delta function into a Lorentzian of width Γ = |Im Σ| and shift it by Re Σ, while transferring some spectral weight (1-Z) to a broad incoherent continuum. ARPES on simple metals shows sharp quasiparticle peaks (Z ~ 0.7-0.9). On strongly correlated materials like cuprate superconductors, the peaks can be broad and Z small, indicating strong deviation from Fermi liquid behavior. ARPES on topological insulators reveals the surface Dirac cone directly.
Question 4 Short Answer
Explain why Feynman diagrams are useful for computing the self-energy, and what the GW approximation captures physically.
Think about your answer, then reveal below.
Model answer: Feynman diagrams provide a systematic graphical expansion of the self-energy in powers of the interaction. Each diagram represents a specific physical process: electron-electron scattering, phonon emission/absorption, repeated scattering events. The diagrammatic approach allows selective summation of important classes of diagrams (e.g., all ring diagrams for screening) rather than computing all diagrams order by order. The GW approximation keeps only the simplest diagram: Σ = iGW, where G is the Green's function and W is the dynamically screened Coulomb interaction. Physically, GW describes an electron propagating through a medium that dynamically screens its Coulomb interaction with other electrons. It captures quasiparticle energy shifts and lifetimes and gives much more accurate band gaps than DFT (typically within 10% of experiment for semiconductors).
The GW approximation is the standard 'beyond-DFT' method for computing quasiparticle band structures. It is the lowest-order diagram in the screened interaction W, which already includes the dominant correlation effect (screening). Higher-order diagrams (vertex corrections) are needed for strongly correlated systems.