Renormalization group approach to time-dependent transport through correlated quantum dots

We introduce a real time version of the functional renormalization group, which allows us to study correlation effects on nonequilibrium transport through quantum dots. Our method is equally capable to address (i) the relaxation out of a nonequilibrium initial state into a (potentially) steady state driven by a bias voltage and (ii) the dynamics governed by an explicitly time-dependent Hamiltonian. All time regimes from transient to asymptotic can be tackled; the only approximation is the consistent truncation of the flow equations at a given order. As an application we investigate the relaxation dynamics of the interacting resonant level model, which describes a fermionic quantum dot dominated by charge fluctuations. Moreover, we study incoherence and relaxation phenomena within the ohmic spin-boson model by mapping the latter to the interacting resonant level model.

Figure 1

Diagramatic representation of the self-energy flow equation. The dot indicates the derivative with respect to $\Lambda$; the slanted line symbolizes the single-scale propagator. Since the flow equation for the self-energy is proportional to a $\delta$ distribution in time we indicate a single time argument $t$ only.

Figure 2

Sketch of the three-site version of the interacting resonant level model.

Figure 3

Comparison of the central-site occupancy between a three-site and a (field-theoretical-like) one-site version of the interacting resonant level model for scaling-limit parameters $\epsilon /\Gamma =V/\Gamma =0.025$, and $\tau /\Gamma =0.025$ (the hybridization in the single-site model reads $\tilde{\Gamma }={\tau }^2/\Gamma$; see the main text for details). The inset illustrates that both models differ only for times $\sim$$1/\Gamma$.

Figure 4

FRG data for the time evolution of the central-site occupancy of the interacting resonant level model depicted in Fig. 2. The parameters read $\tau /\Gamma =0.025$ (local coupling), $\epsilon /T_K=10$ (level position), and $V/T_K=10$ (difference of chemical potentials). The universal equilibrium energy scale $T_K$ is renormalized through the local Coulomb interaction $U$. The arrows indicate the steady-state values obtained by the nonequilibrium steady-state functional RG of Ref. 23.

Figure 5

The same as in Fig. 4 but now showing the current leaving the left reservoir. The negative current found at small times was already observed in Refs. 28 and 31.

Figure 6

Time dependence of the renormalized on-site energy ${\epsilon }^{\mathrm{ren}}\left(t\right)-\epsilon \in \mathbb{R}$ (of the central site 2). The parameters are the same as in Fig. 4: $\tau /\Gamma =0.025$, $\epsilon /T_K=V/T_K=10$, and $U/\Gamma =0.1$. Note the double logarithmic scale. Kinks in the graph are due to the time discretization.

Figure 7

The same as in Fig. 6 but for the renormalized hoppings ${\tau }_{12}^{\mathrm{ren}}-\tau ={\tilde{\Sigma }}_{12}^{\mathrm{ret},\Lambda =0}\in \mathbb{C}$ between sites 1 and 2, and ${\tau }_{23}^{\mathrm{ren}}-\tau ={\tilde{\Sigma }}_{23}^{\mathrm{ret},\Lambda =0}\in \mathbb{C}$ between sites 2 and 3.

Figure 8

Relative difference of the data for the occupancy shown in Fig. 4 and a fit to Eq. (82). Small deviations even in the noninteracting case originate from $\Gamma$ being finite and $t$ being only roughly an order of magnitude larger than $T_K$.

Figure 9

Comparison of FRG and real time RG data for the occupancy ${\overline{n}}_2\left(t\right)$ at $\tau /\Gamma =0.025$, and $\epsilon /T_K=V/T_K=10$ (the parameters of Fig. 4). The curves at $U/\Gamma =0.1$ are almost indistinguishable.

Figure 10

The same as in Fig. 9 but for the current.

Figure 11

Interaction dependence of the power-law exponent extracted from our FRG data by fitting to the RTRG prediction of Eq. (83) at $\epsilon /T_K=30$ (see the main text for details). The solid line shows the leading-order term $2U/\pi$. The error bars refer to a $68\%$ confidence level.

Figure 12

FRG calculation of expectation value $|\langle {\sigma }_z\left(t\right)\rangle |=|2n_2\left(t\right)-1|$ for the ohmic spin-boson model. In absence of a bias voltage, the latter can be mapped to the IRLM, and our fermionic FRG scheme is directly applicable. In terms of the IRLM, the system parameters read $\tau /\Gamma =0.025$ and $\epsilon =V=0$. Note that the $y$ axis is scaled logarithmically and that the times are given with respect to $\Gamma$ instead of $T_K$.

Figure 13

Oscillation frequency governing the relaxation dynamics of the spin-boson model as a function of $U$ for $\tau /\Gamma =0.025$. The functional RG result is compared to the prediction of Eq. (90). The error bars result from a $68\%$ confidence level fitting of the frequency.