F.1.1 Screened Interaction, Polarization, and Vertex Function

The equation (F.12) can be used as a starting point for a diagrammatic expansion. One possible way is to iterate $ G(12)$ in the functional derivative with respect to $ U(3)$, starting from the non-interacting GREEN's function $ G_0$. This procedure is described e.g. in [93], and specifically for the KELDYSH formalism, in [203]. This expansion scheme is based on the non-interacting GREEN's function. In order to avoid the appearance of non-interacting GREEN's functions in the diagrammatic expansion without simultaneously complicating the rules for constructing the diagrams, one has to extend the equations for $ G(12)$. Technically, this extension is based on the repeated change of variables and the consequent application of the chain-rule in the evaluation of the functional derivatives. One usually generates the following additional function Although the expanded set of functions still does not lead to a closed set of equations (an additional function, $ \delta \Sigma/\delta G$, occurs), it allows for a perturbative solution by means of iterating $ \Sigma$ in the derivative $ \delta \Sigma/\delta G$. The formal structure of these equations will turn out to be essentially

\begin{displaymath}\begin{array}{ll} \displaystyle \Sigma \ & \displaystyle = \ ...
...\frac{\delta \Sigma}{\delta G} \ G \ \Gamma \ G \ . \end{array}\end{displaymath} (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]

\begin{displaymath}\begin{array}{ll}\displaystyle \Sigma(12) \ &\displaystyle = ...
...3 \int_\mathrm{C} d4\ W(51)\ G(14)\ \Gamma(425) \ , \end{array}\end{displaymath} (F.14)

where the screened interaction is defined as

\begin{displaymath}\begin{array}{l}\displaystyle W(12)\ = \ \int_\mathrm{C} d3 \...
... \ \frac{\delta U_\mathrm{eff}(1)}{\delta U(3)} \ , \end{array}\end{displaymath} (F.15)

and the vertex function

\begin{displaymath}\begin{array}{ll}\displaystyle \Gamma(123)\ &\displaystyle = ...
...ac{\delta G^{-1}(12)}{\delta U_\mathrm{eff}(3)} \ . \end{array}\end{displaymath} (F.16)

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 functionF.1

\begin{displaymath}\begin{array}{l}\displaystyle \epsilon^{-1}(12)\ = \ \frac{\delta U_\mathrm{eff}(1)}{\delta U(2)} \ , \end{array}\end{displaymath} (F.17)

can be written in terms of the polarization function

\begin{displaymath}\begin{array}{l}\displaystyle \Pi(12)\ = \ -i\hbar\frac{\delta G(11)}{\delta U_\mathrm{eff}(2)} \ , \end{array}\end{displaymath} (F.18)

in the following way

\begin{displaymath}\begin{array}{ll}\displaystyle \frac{\delta U_\mathrm{eff}(1)...
...)\ \frac{\delta U_\mathrm{eff}(4)}{\delta U(2)} \ . \end{array}\end{displaymath} (F.19)

In this way, one obtains

\begin{displaymath}\begin{array}{l}\displaystyle \epsilon^{-1}(12)\ = \ \delta_{...
...thrm{C} d4 \ V(1-3)\ \Pi(34)\ \epsilon^{-1}(42) \ . \end{array}\end{displaymath} (F.20)

and from (F.15)

\begin{displaymath}\begin{array}{l}\displaystyle W(12)\ = \ V(2-1) \ + \ \int_\m...
...d3\ \int_\mathrm{C} d4 \ V(1-3)\ \Pi(34)\ W(42) \ . \end{array}\end{displaymath} (F.21)

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

\begin{displaymath}\begin{array}{ll}\displaystyle \Pi(12)\ & \displaystyle = \ -...
...\int_\mathrm{C} d4 \ G(13) \ \Gamma(342) \ G(41)\ . \end{array}\end{displaymath} (F.22)

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 $ G^{-1}$ in terms of $ G$. One can multiply and integrate both sides of the equation of motion (F.11) by $ G_0^{-1}(32)$ and $ G^{-1}(32)$, where $ G^{-1}_{0}(12) =
(i\hbar\partial_{t_1}+\frac{\hbar^2}{2m}\ensuremath{{\mathbf{\nabla}}}^2_1- U_\mathrm{eff}(1))
\delta_{1,2}$. Finally, one obtains $ G^{-1}(12)=G^{-1}_{0}(12)-\Sigma(12)$, which can be used to rewrite the vertex function (F.16) as

\begin{displaymath}\begin{array}{ll}\displaystyle \Gamma(123)\ &\displaystyle = ...
...)} {\delta G(45)} \ G(46) \ \Gamma(673) \ G(75) \ , \end{array}\end{displaymath} (F.23)

where the relation (F.10) has been used. Contributions proportional to $ \delta \Sigma/\delta G$ are referred to as vertex corrections and describe interaction processes at the two-particle level. M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors