More Advanced Topics in Self-Energy

 

The introductory page presented the self-energy $\Sigma$ as the effective quantity that modifies the reference Green’s function through the Dyson equation,

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

We now examine several deeper aspects of the self-energy: its operator structure, diagrammatic meaning, analytic properties, quasiparticle renormalization, and applications beyond ordinary electronic excitations.

Self-energy is not simply an energy shift

The name “self-energy” may suggest a single numerical correction to the energy of a particle. In general, however, the self-energy is an operator.

In coordinate space, it has two spatial arguments:

$$\Sigma(\mathbf r,\mathbf r’;\omega).$$

Its action on a wave function is

$$\left[ \widehat{\Sigma}(\omega)\psi \right](\mathbf r) = \int d\mathbf r’ \Sigma(\mathbf r,\mathbf r’;\omega) \psi(\mathbf r’).$$

Therefore, the self-energy may be nonlocal: the result at position $\mathbf r$ can depend on the wave function at another position $\mathbf r'$.

In a discrete basis, the self-energy becomes a matrix,

$$\Sigma_{mn}(\omega) = \left\langle m \middle| \widehat{\Sigma}(\omega) \middle| n \right\rangle.$$

Only in a simple one-state or diagonal approximation does it reduce to a scalar function such as

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

Energy-dependent effective operator

The interacting Green’s function may be written as

$$G(\omega) = \left[ \omega I - H_0 - \Sigma(\omega) \right]^{-1}.$$

This suggests the effective operator

$$H_{\mathrm{eff}}(\omega) = H_0+\Sigma(\omega).$$

However, $H_{\mathrm{eff}}$ is not generally an ordinary Hamiltonian. The self-energy may be:

  • energy dependent;
  • nonlocal;
  • complex;
  • non-Hermitian.

The dependence on $\omega$ means that finding the excitation energies usually requires solving a nonlinear problem.

Diagrammatic meaning of the self-energy

Diagrammatically, the self-energy represents interaction processes that modify one-particle propagation.

The self-energy entering the Dyson equation is the proper self-energy, also called the one-particle irreducible self-energy.

A proper self-energy diagram cannot be divided into two disconnected parts by cutting a single internal fermionic propagator line.

The reason for this restriction is that repeated self-energy insertions are already generated by the Dyson equation:

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

Thus, $\Sigma$ contains the individual irreducible interaction processes, while the Dyson equation combines them into repeated propagation sequences.

This avoids counting the same reducible processes twice.

Retarded self-energy

For physical excitation energies and lifetimes, one commonly uses the retarded Green’s function,

$$G^{R}(\mathbf k,\omega) = \frac{1}{ \omega+i\eta - \varepsilon_{\mathbf k}^{0} - \Sigma^{R}(\mathbf k,\omega) }, \qquad
\eta\rightarrow 0^{+}. $$

The retarded self-energy is generally complex:

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

Its real and imaginary parts are not independent. They are connected by Kramers–Kronig relations, which follow from causality.

A strong frequency dependence in one part therefore produces corresponding structure in the other.

Energy renormalization

The real part of the self-energy shifts the energy of the excitation.

The quasiparticle energy approximately satisfies

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

If the reference Hamiltonian already contains an exchange-correlation potential, as in Kohn–Sham density functional theory, that contribution must be treated consistently. A commonly used quasiparticle equation is then

$$E_{n\mathbf k} = \varepsilon_{n\mathbf k}^{\mathrm{KS}} + \left\langle n\mathbf k \middle| \Sigma(E_{n\mathbf k}) - V_{\mathrm{xc}} \middle| n\mathbf k \right\rangle.$$

The subtraction of $V_{\mathrm{xc}}$ prevents the exchange-correlation contribution already present in the reference Hamiltonian from being included twice.

Linewidth and lifetime

The imaginary part of the retarded self-energy describes decay and scattering.

For a well-defined quasiparticle, the linewidth is approximately

$$\Gamma_{\mathbf k} = -2Z_{\mathbf k} \operatorname{Im} \Sigma^{R}(\mathbf k,E_{\mathbf k}).$$

The lifetime is inversely related to the linewidth:

$$\tau_{\mathbf k} \sim \frac{1}{\Gamma_{\mathbf k}}.$$

A small linewidth corresponds to a long-lived excitation and a sharp spectral peak.

A large linewidth corresponds to strong scattering and a short-lived excitation.

Quasiparticle weight

The frequency dependence of the self-energy changes not only the quasiparticle energy but also the strength of its spectral peak.

The quasiparticle renormalization factor is

$$Z_{n\mathbf k} = \left[ 1 - \left. \frac{\partial \operatorname{Re}\Sigma^{R}_{n\mathbf k}(\omega)}
{\partial\omega}
\right|_{\omega=E_{n\mathbf k}} \right]^{-1}.$$

The value $Z_{n\mathbf k}$ measures how much of the interacting excitation retains the character of the original reference particle.

For a weakly interacting quasiparticle,

$$Z_{n\mathbf k} \approx 1. $$

A smaller value indicates stronger many-body dressing and a transfer of spectral weight to additional excitations or satellite structures.

Effective-mass renormalization

The self-energy changes the quasiparticle dispersion and therefore its effective mass.

Both the energy dependence and the momentum dependence of the self-energy can contribute.

Schematically,

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

Differentiating this relation with respect to momentum shows that the renormalized velocity depends on both

$$\frac{\partial\Sigma}{\partial\omega}$$

and

$$\nabla_{\mathbf k}\Sigma.$$

Therefore, it is generally incomplete to associate effective-mass renormalization only with the frequency derivative of the self-energy.

Main physical contributions to the self-energy

The total self-energy may contain contributions from several mechanisms:

$$\Sigma = \Sigma_{\mathrm{e-e}} + \Sigma_{\mathrm{e-ph}} + \Sigma_{\mathrm{imp}} + \Sigma_{\mathrm{spin}} + \cdots.$$

Electron-electron interactions

Coulomb interactions between electrons shift energy levels, renormalize dispersions, and generate finite lifetimes.

In weakly correlated systems, these effects often produce well-defined quasiparticles.

In strongly correlated systems, the same interactions may strongly suppress quasiparticle weight or destroy the quasiparticle picture entirely.

Electron-phonon interactions

Coupling between electrons and lattice vibrations contributes to both energy renormalization and scattering.

It can produce:

  • temperature-dependent linewidths;
  • band-energy shifts;
  • mass enhancement;
  • electrical resistivity;
  • superconducting pairing.

Impurity and disorder scattering

Defects and impurities break translational symmetry and broaden electronic states.

The corresponding imaginary self-energy produces a finite elastic-scattering rate, even at low temperature.

Spin fluctuations

In magnetic and strongly correlated materials, electrons can interact with dynamic spin excitations.

These processes can strongly modify electronic bands and lifetimes and may contribute to non-Fermi-liquid behavior or unconventional superconductivity.

When the quasiparticle picture breaks down

The quasiparticle description requires a reasonably sharp spectral peak.

If

$$\left| \operatorname{Im}\Sigma(\mathbf k,E_{\mathbf k}) \right| $$

becomes too large, the excitation may decay before it can propagate as a well-defined particle.

Strong frequency dependence can also reduce the quasiparticle weight:

$$Z_{\mathbf k}\ll1.$$

In such cases, the spectral function may be dominated by broad continua, Hubbard bands, or other incoherent structures rather than sharp quasiparticle peaks.

Examples include:

  • Mott insulators;
  • non-Fermi liquids;
  • systems near quantum critical points;
  • strongly disordered materials.

Self-energy beyond electrons

The same concept applies to many other excitations.

Phonons

A phonon self-energy shifts vibrational frequencies and produces finite phonon linewidths.

Magnons

A magnon self-energy describes spin-wave renormalization and damping.

Excitons

An exciton self-energy may shift exciton energies, modify binding energies, and produce finite lifetimes through coupling to phonons, photons, or other excitations.

Photons and polaritons

Photon and polariton self-energies describe changes in optical modes caused by coupling to matter and to dissipative environments.

Relation to quantum field theory

The concept of self-energy also appears in relativistic quantum field theory.

For example, an electron can emit and reabsorb virtual photons. These processes modify its propagator and contribute to the difference between bare and observed particle parameters.

The mathematical structure remains the same:

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

The physical interpretation depends on the system being studied.

In this tutorial, the main focus is on electronic excitations in condensed matter.

Historical note: classical self-energy

In classical electrodynamics, the electric field of a charged particle stores energy.

This field energy is sometimes called the classical self-energy.

For an ideal point charge, the corresponding energy diverges, which historically motivated important questions about particle structure and renormalization.

Although related by terminology, this classical concept is distinct from the many-body self-energy operator used in the Dyson equation.

Connection to the GW approximation

The self-energy page introduced the central question:

How can the interaction effects collected in $\Sigma$ be calculated?

The GW approximation provides one widely used answer:

$$\Sigma \approx iGW.$$

Here, the electron propagator $G$ describes particle propagation, while the screened interaction $W$ describes how the Coulomb interaction is modified by the response of the surrounding electronic system.