D.1 Non-Interacting FERMIons

The HAMILTONian for non-interacting electrons (FERMIons) in momentum representation is

$\displaystyle H_{0} \ = \ \sum_{{\bf k}} \xi_{{\bf k}} c^\dagger_{{\bf k}}c_{{\bf k}} \ ,$ (D.1)

where $ \xi_{\bf k}=E_{\bf k}- E_\mathrm{F}$ is the single-particle energy measured with respect to the FERMI energy $ c_{{\bf k}}$ and $ c^\dagger_{{\bf k}}$ are the FERMIon annihilation and creation operators, respectively (Appendix A). The time-evolution of the annihilation operator in the HEISENBERG picture is (Appendix B)

$\displaystyle c_{{\bf k}}(t) \ = \ e^{iH_{0}t/\hbar}\ c_{{\bf k}} \ e^{-iH_{0}t/\hbar} \ ,$ (D.2)

so the operator obeys the equation

$\displaystyle i\hbar \partial_t c_{{\bf k}}(t) \ = \ [c_{{\bf k}}(t),H_{0}] \ = \ \xi_{{\bf k}} c_{{\bf k}}(t)\ ,$ (D.3)

which has the solution

$\displaystyle c_{{\bf k}}(t) \ = \ e^{-i\xi_{{\bf k}}t/\hbar}\ c_{{\bf k}} \ .$ (D.4)

The creation operator for FERMIons is the just the HERMITian conjugate of $ c_{{\bf k}}$, i.e.

$\displaystyle c^\dagger_{{\bf k}}(t) \ = \ e^{i\xi_{{\bf k}}t/\hbar}\ c^\dagger_{{\bf k}} \ .$ (D.5)

The non-interacting real-time GREEN's functions (Section 3.7.1) for FERMIons in momentum representation are now given by

\begin{displaymath}\begin{array}{ll}\displaystyle G_{0}^{<}({\bf k},t;{\bf k'},t...
...{\bf k}}(t'-t)/\hbar} \delta_{{\bf k},{\bf k'}} \ , \end{array}\end{displaymath} (D.6)

where $ n_{\bf k}=\langle c^\dagger_{\bf k} c_{\bf k}\rangle$ is the average occupation number of the state $ {\bf k}$. The GREEN's functions depend only on time differences. One usually Fourier transforms the time difference coordinate, $ t-t'$, to energy

\begin{displaymath}\begin{array}{ll}\displaystyle G_{0}^\mathrm{<}({\bf k},E) \ ...
...\displaystyle = \ \frac{1}{E-\xi_{\bf k}-i\eta} \ , \end{array}\end{displaymath} (D.7)

where $ \eta=0^+$ is a small positive number. Assuming that the particles are in thermal equilibrium one obtains $ n_\mathrm{{\bf k}} = n_\mathrm{F}(\xi_{{\bf
k}})$, where $ n_\mathrm{F}$ is the FERMI-DIRAC distribution function (Appendix C.1). The result (D.7) shows that $ G^{<}$ and $ G^{>}$ provide information about the statistics, such as occupation $ n_{\bf k}$ or un-occupation $ 1-n_{\bf k}$ of the states, and $ G^\mathrm{r}$ and $ G^\mathrm{a}$ provide information about the states regardless of their occupation. The spectral function $ A_{0}({\bf k},E)$ for FERMIons is therefore defined as

\begin{displaymath}\begin{array}{ll} A_{0}({\bf k},E) \ & \displaystyle = \ +i[G...
...pt] & \displaystyle = +2\pi\delta(E-\xi_{\bf k}) \ ,\end{array}\end{displaymath} (D.8)

where the following relation is used

\begin{displaymath}\begin{array}{l}\displaystyle \frac{1}{x\pm i\eta} \ = \ \mat...
...P}\left(\frac{1}{x}\right)\ \mp i\pi\delta(x) \ \ , \end{array}\end{displaymath} (D.9)

where $ \mathcal{P}$ indicates the principal value. Under equilibrium the lesser and greater GREEN's functions can be rewritten as

\begin{displaymath}\begin{array}{l}\displaystyle G^\mathrm{<}_{0}({\bf k},E) \ =...
...k},E) \ = \ i [1-n_\mathrm{F}] A_{0}({\bf k},E) \ . \end{array}\end{displaymath} (D.10)

M. Pourfath: Numerical Study of Quantum Transport in Carbon Nanotube-Based Transistors