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

• the self-energy $\Sigma(12)$, which contains information on both the renormalization of the single-particle energies and the scattering rates.
• the longitudinal polarization function $\Pi(21)$, 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 $W(12)$, which differs from the bare COULOMB potential because of the possibility of single-particle transitions as described by $\Pi$, brought about by charge density fluctuations, and because of the related possibility of collective excitations,
• the vertex function $\Gamma(123)$, which serves to formally complete the set of equations.

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

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

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

where the screened interaction is defined as

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

and the vertex function

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

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

can be written in terms of the polarization function

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

in the following way

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

In this way, one obtains

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

and from (F.15)

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

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

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

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