# Electron Transport in Carbon Nanotubes with Adsorbed Chromium Impurities
## 3 Green's Functions of Electrons and Phonons
Substituting these results into the definition of the thermal Green's function creates the alternate interaction-representation form for the Green's function, given by  
\[G_{a}^{AB}(t,t^{\prime}) = \frac{i}{\hbar}\theta(t^{\prime}-t)\langle[\tilde{A}(t),\tilde{B} (t^{\prime})]\rangle. \tag{32}\]3  
where all time dependence is with respect to the noninteracting Hamiltonian and the trace over all states is with respect to the noninteracting states  
\[G_{a}^{AB}(t,t^{\prime}) = \frac{i}{\hbar}\theta(t^{\prime}-t)\langle[\tilde{A}(t),\tilde{B} (t^{\prime})]\rangle. \tag{32}\]4  
This last result forms the starting point for the perturbative expansion employed here. The diagrammatic method is generated by expanding \(\sigma(\tau)\) in a power series in terms of \(H_{\mathrm{int}}(\tau)\) and then using [PERSON]'s theorem to evaluate the resulting operator averages (since the noninteracting Hamiltonian is quadratic [31]). This technique then generalizes the approach used for the homogeneous system [31]. The denominator in Equation (43) cancels all disconnected diagrams in the expansion, as usual. Therefore, the thermal Green's function are expanded in terms of connected diagrams. Using the standard relations between the spectral representations of thermal and real-time Green's functions [31], we obtain the following Dyson equation for the electronic Green's function in the real frequency domain (hereinafter the dependence on \(r\) is suppressed) [31]:  
\[G_{a}^{AB}(t,t^{\prime}) = \frac{i}{\hbar}\theta(t^{\prime}-t)\langle[\tilde{A}(t),\tilde{B} (t^{\prime})]\rangle. \tag{32}\]5 \[G_{a}^{AB}(t,t^{\prime}) = \frac{i}{\hbar}\theta(t^{\prime}-t)\langle[\tilde{A}(t),\tilde{B} (t^{\prime})]\rangle. \tag{32}\]6 \[G_{a}^{AB}(t,t^{\prime}) = \frac{i}{\hbar}\theta(t^{\prime}-t)\langle[\tilde{A}(t),\tilde{B} (t^{\prime})]\rangle. \tag{32}\]7 \[G_{a}^{AB}(t,t^{\prime}) = \frac{i}{\hbar}\theta(t^{\prime}-t)\langle[\tilde{A}(t),\tilde{B} (t^{\prime})]\rangle. \tag{32}\]8 \[G_{a}^{AB}(t,t^{\prime}) = \frac{i}{\hbar}\theta(t^{\prime}-t)\langle[\tilde{A}(t),\tilde{B} (t^{\prime})]\rangle. \tag{32}\]9 \[\tilde{A}(t)=e^{i\mathrm{H}t/\hbar}Ae^{-i\mathrm{H}t/\hbar} \tag{33}\]0 \[\tilde{A}(t)=e^{i\mathrm{H}t/\hbar}Ae^{-i\mathrm{H}t/\hbar} \tag{33}\]1 \[\tilde{A}(t)=e^{i\mathrm{H}t/\hbar}Ae^{-i\mathrm{H}t/\hbar} \tag{33}\]2 \[\tilde{A}(t)=e^{i\mathrm{H}t/\hbar}Ae^{-i\mathrm{H}t/\hbar} \tag{33}\]3 \[\tilde{A}(t)=e^{i\mathrm{H}t/\hbar}Ae^{-i\mathrm{H}t/\hbar} \tag{33}\]4  
where \(\varepsilon=\hbar\omega\). Here \(G^{aa^{+}}(\varepsilon)\), \(G^{uu}(\varepsilon)\), \(G^{PP}(\varepsilon)\), \(G^{uP}(\varepsilon)\), and \(G^{Pu}(\varepsilon)\) are the real-frequency representations of the single-particle Green's function of the electrons, the coordinate-coordinate, momentum-momentum, coordinate-momentum, and momentum-coordinate Green's functions of the phonons, respectively; \(\Sigma_{eph}(\varepsilon)\), \(\Sigma_{phe}(\varepsilon)\), \(\Sigma_{e\sigma}(\varepsilon)\), and \(\Sigma_{phph}(\varepsilon)\) are the corresponding self-energies (mass operators) for the electron-phonon, phonon-electron, electron-electron, and phonon-phonon interactions. The real-time and real-frequency Green's functions are related by standard Fourier transform relations given by  
\[\tilde{A}(t)=e^{i\mathrm{H}t/\hbar}Ae^{-i\mathrm{H}t/\hbar} \tag{33}\]5  
and  
\[\tilde{A}(t)=e^{i\mathrm{H}t/\hbar}Ae^{-i\mathrm{H}t/\hbar} \tag{33}\]6  
The thermal Green's functions are periodic (bosons) or antiperiodic (fermions) on the interval \(-1/\Theta\leqslant\tau<1/\Theta\), and hence have a Fourier series representation in terms of their Matsubara frequencies, as follows:  
\[\tilde{A}(t)=e^{i\mathrm{H}t/\hbar}Ae^{-i\mathrm{H}t/\hbar} \tag{33}\]7  
and  
\[\tilde{A}(t)=e^{i\mathrm{H}t/\hbar}Ae^{-i\mathrm{H}t/\hbar} \tag{33}\]8  
where the Matsubara frequencies satisfy  
\[\omega_{n}=\left\{ \begin{array}{ll}2n\pi\,\Theta&\mbox{for Bose particles}\\ (2n+1)\pi\,\Theta&\mbox{for Fermi particles}\end{array} \right.