Quasiparticles
A quasiparticle is a long-lived excitation of an interacting system that behaves approximately like a particle with renormalized properties. It is not a bare electron: the surrounding medium responds, and that dressing changes the excitation energy, spectral weight, velocity, and lifetime.
From Dyson’s equation to a pole
For a single band, the retarded Green’s function can be written as
$$ G^R(\mathbf k,\omega)= \frac{1}{\omega+\mu-\epsilon_{\mathbf k}-\Sigma^R(\mathbf k,\omega)}. $$
The non-interacting pole lies at $\epsilon_{\mathbf k}-\mu$. Interactions replace it with the solution of the quasiparticle equation
$$ E_{\mathbf k}+\mu-\epsilon_{\mathbf k} -\operatorname{Re}\Sigma^R(\mathbf k,E_{\mathbf k})=0. $$
Because the self-energy depends on frequency, this equation is nonlinear and may have more than one solution. A recognizable quasiparticle requires one solution to carry appreciable spectral weight and have a width small compared with its characteristic energy scale.
Renormalization factor
Expand the real part of the self-energy near the quasiparticle energy. The coherent pole has residue
$$ Z_{\mathbf k}=\left[ 1-\left.\frac{\partial\operatorname{Re}\Sigma^R(\mathbf k,\omega)}{\partial\omega} \right|{\omega=E{\mathbf k}} \right]^{-1}. $$
$Z_{\mathbf k}$ measures the overlap between the bare-particle state and the dressed excitation. Interactions transfer the missing weight $1-Z_{\mathbf k}$ into incoherent backgrounds and satellite features. In a matrix problem, the derivative is evaluated in the relevant quasiparticle state rather than treated as a scalar.
Spectral function and lifetime
The quantity measured by charged-excitation spectroscopies is related to the spectral function
$$ A(\mathbf k,\omega)=-\frac{1}{\pi}\operatorname{Im}G^R(\mathbf k,\omega). $$
Near a well-isolated pole, the coherent contribution is approximately Lorentzian:
$$ A_{\mathrm{qp}}(\mathbf k,\omega)\approx \frac{Z_{\mathbf k}}{\pi} \frac{\Gamma_{\mathbf k}/2} {(\omega-E_{\mathbf k})^2+(\Gamma_{\mathbf k}/2)^2}, $$
with linewidth
$$ \Gamma_{\mathbf k}\approx -2Z_{\mathbf k}\operatorname{Im}\Sigma^R(\mathbf k,E_{\mathbf k}), \qquad \tau_{\mathbf k}\approx\frac{\hbar}{\Gamma_{\mathbf k}}. $$
Thus the two parts of the self-energy have complementary roles: $\operatorname{Re}\Sigma$ shifts the excitation, while $\operatorname{Im}\Sigma$ gives decay and broadening.
What a spectrum is telling you
| Feature | Interpretation |
|---|---|
| Sharp peak with large $Z$ | Long-lived, particle-like excitation |
| Broad peak | Short lifetime or strong scattering |
| Satellite peak | Coupling to another excitation, such as a plasmon |
| Strong incoherent continuum | Spectral weight not captured by a single quasiparticle |
Photoemission probes electron removal; inverse photoemission probes electron addition. Neither experiment measures a Kohn–Sham eigenvalue directly. In Green’s-function methods, quasiparticle energies emerge from the interacting propagator and its self-energy.
When the picture breaks down
The quasiparticle approximation is controlled only when $\Gamma_{\mathbf k}$ is small and the self-energy varies smoothly near the pole. It becomes unreliable when peaks strongly overlap, $Z$ becomes very small, satellites compete with the main peak, or the Green’s function has no isolated pole. Strongly correlated metals, systems near quantum criticality, and fractionalized phases can require a description centered on the full spectral function rather than individual quasiparticles.
Connection to GW
The GW approximation constructs $\Sigma\approx iGW$ from a propagator and a dynamically screened interaction. Its most common use is to correct charged-excitation energies, but the same frequency-dependent self-energy also contains spectral weights, lifetimes, and satellites. A one-shot $G_0W_0$ calculation evaluates these corrections from a fixed reference; self-consistent GW updates the propagator and screening together.