Tunneling-induced renormalization in interacting quantum dots

We analyze tunneling-induced quantum fluctuations in a single-level quantum dot with arbitrarily strong on-site Coulomb interaction, generating cotunneling processes and renormalizing system parameters. For a perturbative analysis of these quantum fluctuations, we remove off-shell parts of the Hamiltonian via a canonical transformation. We find that the tunnel couplings for the transitions connecting empty and single occupation and connecting single and double occupation of the dot renormalize with the same magnitude but with opposite signs. This has an important impact on the shape of the renormalization extracted, for example, from the conductance. Finally, we verify the compatibility of our results with a systematic second-order perturbation expansion of the linear conductance performed within a diagrammatic real-time approach.

Figure 1

Level renormalization $\delta \epsilon$ as a function of the bare level position $\epsilon$ in units of $\Gamma$, for zero bias voltage. We furthermore choose $k_{\mathrm{B}}T=2\Gamma$ (full line) respectively $k_{\mathrm{B}}T=0$ (dashed line) and $U=30\Gamma$.

Figure 2

Renormalization of the tunnel-coupling strengths for the two transitions between empty and singly occupied ($\delta {\Gamma }_{\epsilon }$, black) and between singly and doubly occupied dot ($\delta {\Gamma }_{\epsilon +U}$, red) as functions of the level position $\epsilon$ in units of $\Gamma$, for zero bias voltage. We furthermore chose $k_{\mathrm{B}}T=2\Gamma$ (full lines) respectively $k_{\mathrm{B}}T=0$ (dashed lines) and $U=30\Gamma$.

Figure 3

(a) Dimensionless conductance as a function of the level position in units of $\Gamma$ in first order (black, dashed-dotted line), including second-order corrections due to real cotunneling (red, dashed line), and including all second-order corrections (blue, full line). (b) Separate second-order corrections: real cotunelling (pink, dashed-dotted line), level renormalization (green, dashed line), and $\Gamma$ renormalization (violet, full line). We choose ${\Gamma }_{\mathrm{L}}={\Gamma }_{\mathrm{R}}$, $k_{\mathrm{B}}T=2\Gamma$, and $U=30\Gamma$.

Figure 4

Combined renormalization of the line width $\delta \Gamma$ as function of the level position $\epsilon$ in units of $\Gamma$, for zero bias voltage. We choose $k_{\mathrm{B}}T=2\Gamma$ and $U=30\Gamma$.