F.1.1 Screened Interaction, Polarization, and Vertex Function

- the self-energy , which contains information on both the renormalization of the single-particle energies and the scattering rates.
- the longitudinal polarization function , which describes the possible single-particle transitions as a result of a longitudinal electric field (which can either be an external field or the result of charge density fluctuations in the system),
- the screened COULOMB potential , which differs from the bare COULOMB potential because of the possibility of single-particle transitions as described by , brought about by charge density fluctuations, and because of the related possibility of collective excitations,
- the vertex function , which serves to formally complete the set of equations.

(F.13) |

By applying the chain rule for functional derivatives, one can introduce the derivative with respect to the effective potential. This allows one to write the self-energy (F.12) as [203]

where the

and the vertex function

Using the definition of the effective potential (F.7) together with (F.9) and the chain rule, the screened COULOMB potential, or equivalently, the inverse dielectric function

can be written in terms of the polarization function

in the following way

In this way, one obtains

and from (F.15)

By using the relation (F.10) one can express the polarization in terms of the vertex function

The system of equations defining the self-energy is closed by the equation for the vertex functions. For that purpose one needs an explicit expression for in terms of . One can multiply and integrate both sides of the equation of motion (F.11) by and , where . Finally, one obtains , which can be used to rewrite the vertex function (F.16) as

where the relation (F.10) has been used. Contributions proportional to are referred to as