Dyson Equations and Self-Energies

The exact self-energy is defined by Dyson’s equation, $G^{-1}=G_0^{-1}-\Sigma$ . It collects one-particle-irreducible insertions relative to a declared reference propagator. Its real part shifts dispersions, its imaginary part contributes damping, and its frequency derivative changes pole residue; none of these pieces alone fixes the incoherent spectral background or guarantees a conserving approximation. The underlying irreducible-insertion reorganization is given in Dyson 1949, pp. 491–494.

Required background. Quantum-Matter Correlators and Observable Conventions fixes the retarded convention. The Generating Functional and Connected Correlators and Cumulants supply the diagrammatic definition.

Helpful background. Lehmann Representations and Spectral Functions in Matter supplies exact positivity and normalization checks.

Dyson’s equation

In operator or matrix notation,

G=G_0+G_0\Sigma G, \qquad G^{-1}=G_0^{-1}-\Sigma.

The split is not unique: changing $G_0$ moves one-body terms between reference and self-energy. Physical poles and observables are unchanged only when the transformation is carried through consistently.

For a translationally invariant normal state,

G^R(\mathbf k,\omega) =\frac1{\omega-\xi_{\mathbf k}-\Sigma^R(\mathbf k,\omega)}, \qquad \xi_{\mathbf k}=\varepsilon_{\mathbf k}-\mu.

Causality makes $\Sigma^R$ analytic in the upper half-plane. For a stable diagonal fermion channel, $\operatorname{Im}\Sigma^R\le0$ on the real axis. A sign violation may indicate an inconsistent continuation or an approximation that has lost spectral positivity.

In an orbital, spin, or Nambu basis, all quantities are matrices and poles follow

\det[G_0^{-1}(\mathbf k,z)-\Sigma(\mathbf k,z)]=0.

Matrix residues are projectors or weight matrices, not a single scalar $Z$ .

Pole, residue, and damping

Suppose an isolated narrow normal-state pole exists near real energy $E_{\mathbf k}$ . Define it by

E_{\mathbf k}-\xi_{\mathbf k} -\operatorname{Re}\Sigma^R(\mathbf k,E_{\mathbf k})=0.

Expanding the denominator gives

Z_{\mathbf k} =\left[ 1-\partial_\omega\operatorname{Re}\Sigma^R(\mathbf k,\omega) \big|_{E_{\mathbf k}} \right]^{-1}, \qquad \gamma_{\mathbf k} =-Z_{\mathbf k}\operatorname{Im}\Sigma^R(\mathbf k,E_{\mathbf k}).

Then

G^R\simeq\frac{Z_{\mathbf k}} {\omega-E_{\mathbf k}+i\gamma_{\mathbf k}} +G^R_{\rm inc}.

The Lorentzian full width at half maximum is $2\gamma_{\mathbf k}$ . The retarded amplitude decays as $e^{-\gamma t}$ ; conventions that call $2\gamma$ the decay rate must be translated explicitly. The pole carries weight $Z$ , while $G_{\rm inc}$ carries the rest required by the spectral sum rule.

The group velocity is

\mathbf v^*_{\mathbf k} =Z_{\mathbf k} \left[\boldsymbol\nabla_{\mathbf k}\xi_{\mathbf k} +\boldsymbol\nabla_{\mathbf k}\operatorname{Re}\Sigma^R \right]_{\omega=E_{\mathbf k}}.

Thus residue and effective mass are distinct: momentum dependence of $\Sigma$ contributes to velocity even when $Z$ is known.

Spectral reconstruction and checks

The spectral function is

A(\mathbf k,\omega) =\frac{-2\operatorname{Im}\Sigma^R} {[\omega-\xi_{\mathbf k}-\operatorname{Re}\Sigma^R]^2 +[\operatorname{Im}\Sigma^R]^2},

when no additional infinitesimal or matrix structure is relevant. A narrow maximum can arise without an isolated pole if a branch edge or rapidly varying numerator is present. The pole must be found in the complex plane or by a controlled narrow-width expansion.

Kramers–Kronig relations connect real and imaginary parts. Adding a constant imaginary width over all frequencies produces a logarithmically ill-behaved real part unless ultraviolet behavior is supplied. Any phenomenological self-energy should therefore state its frequency window and completion.

At large frequency, $G(z)\sim1/z$ . If an approximate self-energy changes this coefficient, it violates canonical normalization. Higher inverse powers are checked against spectral moments.

Extracting a self-energy

Given a measured or computed $G$ and a chosen $G_0$ ,

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

This inversion amplifies noise near zeros of $G$ and inherits uncertainty in band structure, chemical potential, matrix elements, and analytic continuation. A fitted $\Sigma$ is model dependent unless these inputs and their covariance are propagated.

In broken-symmetry Nambu space, normal and anomalous self-energies depend on spinor convention and gauge. Observable poles are invariant after the entire matrix and source vertices are transformed; an anomalous component by itself is not a measured condensate.

Common pitfalls

Calling $\operatorname{Re}\Sigma$ the dispersion. The pole solves an implicit equation and includes the reference dispersion. Its derivative also enters $Z$ and velocity.

Equating $Z$ with effective mass. They coincide only under additional assumptions such as a momentum-independent self-energy in a simple band.

Inferring conservation from Dyson self-consistency. A self-consistent $G$ still needs a compatible response vertex. Conservation is tested at the two-particle level.

Exercises

Linear self-energy

Let $\Sigma^R(\omega)=\Sigma_0+(1-Z_0^{-1})\omega-i\Gamma_0$ . Find the pole and damping.

Solution

The denominator is $Z_0^{-1}\omega-\xi-\Sigma_0+i\Gamma_0$ . Hence

E=Z_0(\xi+\Sigma_0), \qquad \gamma=Z_0\Gamma_0,

and the pole residue is $Z_0$ . Positivity requires $\Gamma_0\ge0$ in this convention.

Separate residue and mass

For $\Sigma(\mathbf k,\omega)=a\omega+b\mathbf k^2/(2m)$ near a parabolic band, find $Z$ and $m^*$ .