Interpolative approach for electron-electron and electron-phonon interactions: From the Kondo to the polaronic regime

We present a theoretical approach to determine the electronic properties of nanoscale systems exhibiting strong electron-electron and electron-phonon interactions and coupled to metallic electrodes. This approach is based on an interpolative ansatz for the electronic self-energy which becomes exact both in the limit of weak and strong coupling to the electrodes. The method provides a generalization of previous interpolative schemes which have been applied to the purely electronic case extensively. As a test case we consider the single level Anderson-Holstein model. The results obtained with the interpolative ansatz are in good agreement with existing data from numerical renormalization group calculations. We also check our results by considering the case of the electrodes represented by a few discrete levels which can be diagonalized exactly. The approximation describes properly the transition from the Kondo regime where electron-electron interactions dominate to the polaronic case characterized by a strong electron-phonon interaction.

Figure 1

Lower order diagrams contributing to the proper self-energy. Diagrams (a) and (b) correspond to the spinless case, diagrams (c)–(f) to the spin-dependent Anderson-Holstein model.

Figure 2

(Color online) Localized level spectral density for a discrete spinless Anderson-Holstein model with an electrode consisting of a three atom tight-binding chain for different values of the electron-phonon coupling parameter $\lambda$ with ${\omega }_0=1$. From upper to lower panel, $\lambda =0.6,1,1.4$. The (blue) continuous line corresponds to the interpolative solution while the (black) dotted one to the numerical diagonalization results. The diagonal energy levels in the chain are taken as zero and the first-neighbor hopping parameter is fixed at $t=1$. The hopping parameter between the dot level and the chain is $V=0.5$ and the level position is the one corresponding to an $e\text{−}h$ symmetric case. A small imaginary part, $\eta =0.05$, is included in both spectral densities to facilitate the comparison.

Figure 3

(Color online) Localized level spectral density for the continuous spinless Anderson-Holstein model for different values of the electron-phonon coupling parameter for an $e\text{−}h$ symmetric case with $\Gamma =0.25$ and ${\omega }_0=1$ (from upper to lower panel, $\lambda =0.3,1,1.4$). The (blue) continuous line corresponds to the interpolative solution and the (black) dotted line in the upper panel to the second-order self-energy approximation.

Figure 4

(Color online) Localized level spectral density for a discrete spinless Anderson-Holstein model with an electrode consisting of a three-atom tight-binding chain for an asymmetric electron-hole case. The (blue) continuous line corresponds to the interpolative solution while the (black) dotted one to the numerical diagonalization results. The renormalized level position is ${\widetilde{ϵ}}_0=-0.2$ with $\lambda =1$ and ${\omega }_0=1$. The diagonal energy levels in the chain are taken as zero and the first-neighbor hopping parameter in the chain is fixed at $t=1$. The hopping parameter between the dot level and the chain is $V=0.25$. A small imaginary part, $\eta =0.05$, is included in both spectral densities to facilitate the comparison.

Figure 5

(Color online) Localized level spectral density for the continuous spinless Anderson-Holstein model for an electron-hole asymmetric case. The parameter values are $\lambda =1$, $\Gamma =0.25$, ${\omega }_0=1$, and ${\widetilde{ϵ}}_0=-0.25$. The (blue) continuous line corresponds to the interpolative solution and the (black) dashed line to the noninteracting case with ${ϵ}_0={\widetilde{ϵ}}_0$.

Figure 6

(Color online) Localized level spectral density for a discrete Anderson-Holstein model with the electrode simulated by a single discrete level (taken as the zero of energy) for different values of the electron-phonon coupling $\lambda$ with ${\omega }_0=1$. From upper to lower panel, $\lambda =0.9,2.6$. The (blue) continuous line corresponds to the interpolative solution while the (black) dotted one to the numerical diagonalization results. The hopping parameter between the dot level and the chain is $V=0.25$ and the dot level position is the one corresponding to an $e\text{−}h$ symmetric case. A small imaginary part, $\eta =0.1$, is included in both spectral densities to facilitate the comparison. The Coulomb interaction parameter is $U=8$.

Figure 7

Localized level spectral density of the continuous Anderson-Holstein model for a fixed value of the Coulomb interaction $U$ and for different values of the electron-phonon coupling constant. The magnitude of the parameters has been chosen as to allow a comparison with existing NRG calculations. $\Gamma =0.0159$, ${\omega }_0=0.05$, $U=0.1$, and $\lambda =0,0.03,0.07,0.09$ (corresponding, respectively to continuous, dashed, dotted, and dash-dotted lines). The level position corresponds to an electron-hole symmetric case.