# Electron Transport in Carbon Nanotubes with Adsorbed Chromium Impurities
## 3 Green's Functions of Electrons and Phonons
\tag{50}\]Note that the thermal Green's functions are equal to the retarded Green's functions evaluated at the Matsubara frequencies for positive Matsubara frequencies and are equal to the advanced Green's functions evaluated at the Matsubara frequencies for negative Matsubara frequencies. The electronic Green's functions are infinite matrices with indices given by the lattice site \(n\), the basis site \(i\), and the other quantum numbers \(\gamma\). Similarly, the phonon Green's functions are also infinite matrices with the same lattice and basis site dependence plus a dependence on the spatial coordinate direction \(\alpha\). This produces some simple equations for the noninteracting Green's functions, namely [31]  
\[G_{0}^{aa^{+}}(\epsilon)=[\epsilon-H_{0}^{(1)}]^{-1} \tag{51}\]  
with  
\[H_{0}^{(1)}=\left\|H_{ni^{\prime},n^{\prime}i^{\prime}\gamma^{\prime}}^{(0)}\right\| \tag{52}\]  
\[G_{0}^{uu}(\epsilon)=\left[\omega^{2}M^{(0)}-\Phi^{(0)}\right]^{-1} \tag{53}\]  
\[\Phi^{(0)}=\left\|\Phi_{ni\alpha,n^{\prime}i^{\prime}\alpha^{\prime}}^{(0)}\right\| \tag{54}\]  
and  
\[M^{(0)}=\left\|M_{i}\delta_{nn^{\prime}}\delta_{ii^{\prime}}\delta_{aa^{ \prime}}\right\|. \tag{55}\]  
Here, the double lines denote a matrix. When the perturbations are small, given by  
\[\frac{\left(\frac{\epsilon^{2}}{\hbar^{2}}\Delta M+\Delta\Phi+\Sigma_{phe}( \epsilon)+\Sigma_{phph}(\epsilon)\right)_{nia,n^{\prime}i^{\prime}\alpha^{ \prime}}}{\Phi_{nia,n^{\prime}i^{\prime}\alpha^{\prime}}^{(0)}}\ll 1, \tag{56}\]  
then the solution of the system of equations in Equation (45) becomes  
\[G^{aa^{+}}(\epsilon) = \left[\left[G_{0}^{aa^{+}}(\epsilon)\right]^{-1}-\left(w+\Sigma_{ eph}(\epsilon)+\Sigma_{ae}(\epsilon)\right)\right]^{-1}\] \[G^{uu}(\epsilon) = \left[\left[G_{0}^{uu}(\epsilon)\right]^{-1}\right.\] \[- \left.\left(\frac{\epsilon^{2}}{\hbar^{2}}\Delta M+\Delta\Phi+ \Sigma_{phe}(\epsilon)+\Sigma_{phph}(\epsilon)\right)\right]^{-1}\] \[G^{PP}(\epsilon) = \frac{\epsilon^{2}}{\hbar^{2}}(M^{(0)})^{2}\,G^{uu}(\epsilon) \tag{57}\]  
where  
\[H_{0}^{(1)}=\left\|H_{ni^{\prime},n^{\prime}i^{\prime}\gamma^{\prime}}^{(0)}\right\| \tag{52}\]0  
These solutions only include the first-order corrections due to the small terms in Equation (56). Ref. [31] summarizes the explicit formulas for the corresponding self-energies, which we do not reproduce here. The electronic self-energy due to the electron-phonon interaction \(\Sigma_{eph}(\tau,\tau^{\prime})\) is described by the diagram in \[H_{0}^{(1)}=\left\|H_{ni^{\prime},n^{\prime}i^{\prime}\gamma^{\prime}}^{(0)}\right\| \tag{52}\]1 The solid lines correspond to electronic propagators \(G_{ni\gamma,n^{\prime}i^{\prime}\tau^{\prime}\gamma^{\prime}}^{(\tau,\tau^{ \prime})}\) and the dashed lines correspond to phonon propagators \(G_{ni\alpha,n^{\prime}i^{\prime}\pi^{\prime}\alpha^{\prime}}^{(\tau,\tau^{ \prime})}\). The vertex correction \(\Gamma^{n_{2}i_{2}i_{1}\gamma_{1}}_{ni\gamma_{1}i_{1}\gamma_{1}}(\tau_{2},\tau, \tau_{1})\) is given by the diagrams in \[H_{0}^{(1)}=\left\|H_{ni^{\prime},n^{\prime}i^{\prime}\gamma^{\prime}}^{(0)}\right\| \tag{52}\]2 Note that the unshaded triangle corresponds to the equation  
\[H_{0}^{(1)}=\left\|H_{ni^{\prime},n^{\prime}i^{\prime}\gamma^{\prime}}^{(0)}\right\| \tag{52}\]3  
In Figures 1 and 2, the internal summations for \(\tilde{n}\gamma\) imply both a summation over \(ni\gamma\) and an integration over the internal time \(\tau\). Each diagram has an overall sign determined by \((-1)^{n+F}\), where \(N\) is the order of the diagram (number of vertices \(\Gamma_{0}\)), and \(F\) is the number of electronic Green's function lines \(G^{aa^{+}}\).