Self-Energy

In the previous section, the self-energy $\Sigma$ appeared in the Dyson equation,

$$G^{-1} = G_0^{-1} - \Sigma.$$

The reference Green’s function $G_0$ describes propagation in a chosen reference system, while the full Green’s function $G$ describes propagation after interaction effects have been included.

The role of the self-energy is to collect these interaction effects into a single effective quantity.

Physical meaning

As an electron propagates through a material, it interacts with its environment.

For example, it may:

  • interact with other electrons;
  • polarize the surrounding electron density;
  • emit or absorb phonons;
  • scatter from defects or impurities;
  • interact with collective excitations.

These processes modify the propagation of the electron.

Instead of describing every interaction event separately, many-body theory collects their combined effect into the self-energy

$$\Sigma.$$

The self-energy is therefore not simply an additional numerical energy. It is an operator that modifies the Green’s function.

As in the previous section, operator hats and the explicit identity operator are omitted for simplicity.

Diagrammatic interpretation

In a Feynman diagram, the self-energy is represented by an insertion on a fermionic propagator line.

The insertion summarizes an interaction process that changes the propagation of the particle.

The Dyson equation,

$$G = G_0 + G_0\Sigma G,$$

then states that the full propagator consists of the reference propagator plus propagation containing a self-energy process.

Repeated self-energy insertions are generated automatically when the Dyson equation is solved:

$$G = G_0 + G_0\Sigma G_0 + G_0\Sigma G_0\Sigma G_0 + \cdots.$$

Thus, $\Sigma$ describes the interaction process itself, while the Dyson equation combines such processes into repeated sequences.

Energy and momentum dependence

In a translationally invariant system, the self-energy is commonly written as

$$\Sigma(\mathbf k,\omega), $$

where $\mathbf k$ is momentum and $\omega$ is frequency or energy.

The interacting Green’s function can then be written schematically as

$$G(\mathbf k,\omega) = \frac{1}{ \omega-\varepsilon_{\mathbf k}^{0}-\Sigma(\mathbf k,\omega)}.$$

Here,

$$\varepsilon_{\mathbf k}^{0}$$

is the reference particle energy.

This expression shows that the self-energy changes the location and shape of the Green’s-function pole.

Real and imaginary parts

The self-energy is generally complex:

$$\Sigma(\mathbf k,\omega) = \operatorname{Re}\Sigma(\mathbf k,\omega) + i\operatorname{Im}\Sigma(\mathbf k,\omega).$$

The two parts have different physical meanings.

The real part, $\operatorname{Re}\Sigma$, shifts and renormalizes the excitation energy.

The imaginary part, $\operatorname{Im}\Sigma$, describes scattering and decay processes. It gives the excitation a finite linewidth and therefore a finite lifetime.

Schematically,

$$\text{energy shift} \longleftrightarrow \operatorname{Re}\Sigma,$$

while

$$\text{linewidth and lifetime} \longleftrightarrow \operatorname{Im}\Sigma.$$

Quasiparticle energy

If the interacting excitation remains sufficiently well defined, it can be described as a quasiparticle.

Its energy approximately satisfies

$$E_{\mathbf k} = \varepsilon_{\mathbf k}^{0} + \operatorname{Re} \Sigma(\mathbf k,E_{\mathbf k}).$$

Compared with the reference energy $\varepsilon_{\mathbf k}^{0}$, the quasiparticle energy is shifted by the real part of the self-energy.

A weak imaginary part corresponds to a sharp, long-lived excitation.

A large imaginary part produces a broad, short-lived excitation.

Spectral function

The measurable excitation spectrum is described by the spectral function

$$A(\mathbf k,\omega) = -\frac{1}{\pi} \operatorname{Im} G^{R}(\mathbf k,\omega).$$

Sharp peaks of the spectral function correspond to long-lived quasiparticle excitations.

Broad peaks correspond to states with shorter lifetimes and stronger scattering.

Because the full Green’s function depends on the self-energy, the structure of

$$A(\mathbf k,\omega)$$

is directly controlled by the frequency dependence and imaginary part of $\Sigma$.

The Dyson equation therefore connects microscopic interaction processes to experimentally observable quantities such as:

  • quasiparticle energies;
  • band renormalization;
  • spectral linewidths;
  • satellite peaks;
  • excitation lifetimes.

Sources of self-energy

Different physical mechanisms contribute to the total self-energy.

Examples include:

  • electron-electron interactions;
  • electron-phonon coupling;
  • impurity and disorder scattering;
  • spin fluctuations;
  • interactions with collective electronic excitations.

The total self-energy may be viewed schematically as

$$\Sigma = \Sigma_{\mathrm{electron-electron}} + \Sigma_{\mathrm{electron-phonon}} + \Sigma_{\mathrm{impurity}} + \cdots.$$

Different approximations retain different contributions.

Approximations for the self-energy

Different methods approximate different parts of the self-energy.

Hartree–Fock

Hartree–Fock produces a static exchange self-energy.

It includes exchange exactly at the mean-field level but does not describe dynamical screening or finite quasiparticle lifetimes.

GW approximation

In the GW approximation,

$$\Sigma \approx iGW,$$

where $G$ is the Green’s function and $W$ is the screened Coulomb interaction.

This approximation is widely used for quasiparticle band-structure corrections in weakly and moderately correlated materials.

Dynamical mean-field theory

In dynamical mean-field theory, the self-energy is treated as frequency dependent but approximately local:

$$\Sigma(\mathbf k,\omega) \approx \Sigma(\omega).$$

This approach is especially useful for strongly correlated systems in which local electronic interactions dominate.

The next section explains how this expression is constructed and how screening enters the theory.

More advanced treatment

For a more detailed discussion of nonlocal and matrix self-energies, proper self-energy diagrams, retarded self-energies, quasiparticle weight, effective-mass renormalization, linewidths, Kramers–Kronig relations, and cases where the quasiparticle picture breaks down, see: More advanced topics in self-energy

Next: Part IV — The GW Approximation