Interacting resonant-level model in nonequilibrium: Finite-temperature effects

We study the steady-state properties as well as the relaxation dynamics of the nonequilibrium interacting resonant-level model at finite temperatures. It constitutes the prototype model of a correlated charge fluctuating quantum dot. The two reservoirs are held at different chemical potentials—the difference being the bias voltage—and different temperatures; we discuss the transport through as well as the occupancy of the single level dot. First, we show analytically that in the steady state the reservoir temperatures in competition with the other energy scales act as infrared cutoffs. This is rather intuitive but, depending on the parameter regime under consideration, leads to a surprisingly rich variety of power laws in the current as a function of the temperatures and the bias voltage with different interaction dependent exponents. Next we clarify how finite reservoir temperatures affect the dynamics. They allow one to tune the interplay of the two frequencies characterizing the oscillatory part of the time evolution of the model at zero temperature. For the exponentially decaying part we disentangle the contributions of the level-lead hybridization and the temperatures to the decay rates. We identify a coherent-to-incoherent transition in the long-time dynamics as the temperature is raised. It occurs at an interaction dependent critical temperature. Finally, taking different temperatures in the reservoirs we discuss the relaxation dynamics of a temperature gradient driven current.

Figure 1

Renormalized linewidth ${\Theta }^{\Lambda =0}$ as a function of temperature $T$. Parameters are chosen as $\epsilon /T_K=V/T_K=0$, ${\tau }_{12}/\Gamma ={\tau }_{23}/\Gamma =0.0025$, $T=T_L=T_R$, and $U/\Gamma =0.1$. Since left and right leads are chosen symmetrically ${\Theta }_{12}={\Theta }_{23}=\Theta$ holds. Notice the double logarithmic scale, which indicates the power-law behavior for large $T$. The inset shows the replacement of the trigamma function used in approximation 2.

Figure 2

Time dependence of the renormalized hoppings ${\tau }_{12}^{\Lambda =0}-\tau ={\Sigma }_{12}^{\mathrm{ret},\Lambda =0}\in \mathbb{C}$ between sites 1 and 2 and of ${\tau }_{23}^{\Lambda =0}-\tau ={\Sigma }_{23}^{\mathrm{ret},\Lambda =0}\in \mathbb{C}$ between sites 2 and 3. The parameters are ${\tau }_{12}={\tau }_{23}=\tau$ with $\tau /\Gamma =0.0025$, $\epsilon /T_K=V/T_K=10$, and $U/\Gamma =0.2$. The insets show a zoom into the regime where one can most clearly distinguish the temperature's effect on the oscillations' amplitude.

Figure 3

FRG data for the time evolution of the central-site occupancy. The parameters are the same as in Fig. 2 and symmetric temperatures $T_{\alpha }=T$. The arrows right of the graph indicate the steady-state values obtained by the nonequilibrium steady-state functional RG discussed earlier.

Figure 4

The same as in Fig. 3 but for the current leaving the left reservoir.

Figure 5

The same as Fig. 3 with $U/\Gamma =0.2$, but featuring a temperature gradient $T_L=T+\Delta T/2$ and $T_R=T-\Delta T/2$ for $T=0.5T_K$. We find how the temperature gradient pronounces the frequency $|\epsilon +V/2|$ belonging to the colder reservoir. This can also be found in the inset, which shows the numerical Laplace transform with the positions of the frequencies $|\epsilon \pm V/2|$ indicated by arrows.

Figure 6

Time evolution for the temperature gradient induced current leaving the left reservoir $J_L\left(t\right)$. The parameters read $\tau /\Gamma =0.0025$, $\epsilon /T_K=10$, and $V/T_K=0$. We choose $T_L=T_K$ and $T_R=0$ (choosing $0<T_R\ll T_L$ does not alter the qualitative behavior).