Finite-frequency thermoelectric response in strongly correlated quantum dots

We investigate the finite-frequency thermal transport through a quantum dot subject to strong interactions by providing an exact, nonperturbative formalism that allows us to carry out a systematic analysis of the thermopower at any frequency. Special emphasis is put on the dc and high-frequency limits. We demonstrate that, in the Kondo regime, the ac thermopower is characterized by a universal function that we determine numerically.

Figure 1

Sketch of the real and imaginary parts of the universal function $s(\omega /T_K,T/T_K)$ in the Kondo regime, at a fixed temperature $T\ll T_K$. The coefficients $a^{\prime }$ and $a^{\prime \prime }$ depend on temperature as $\sim 1/T^2$. See also Eq. (21) for the functional dependence of $s$.

Figure 2

Sketch of the quantum dot that is coupled to two external leads which are assumed to be at different temperatures $T\pm \delta T\left(t\right)/2$.

Figure 3

Density plots for the real and imaginary parts of the thermopower in the $(T,\omega )$ plane: (a), (b) Kondo regime, (c), (d) mixed valence regime, and (e), (f) empty orbital regime. Panels (g) and (h) represent the thermopower at $T=0.1\Gamma$ for positive $\omega$ along the dashed lines in the density plots. The darker lines are the locations of the zeros and mark the positions where the thermopower changes sign. The $(\pm )$ symbols indicate the signs of $S\left(\omega \right)$. The NRG parameters used are $U/\Gamma =10$, ${\varepsilon }_d/\Gamma =-4(n=0.985)$, ${\varepsilon }_d/\Gamma =-0.2(n=0.5)$, and ${\varepsilon }_d/\Gamma =5(n=0.04)$.

Figure 4

(a) The dc thermopower $S_0=S(\omega =0)$ as function of temperature in different regimes. (b) Temperature dependence of $S^{*}=S(\omega \gg D)$.

Figure 5

Universal scaling functions for a filling $〈n〉=0.95$ for a fixed temperature $T=0.02T_K$. In the inset, we represent the $s_0$ universal function.

Figure 6

The imaginary part of the dynamical coefficients $L_{ij}\left(\omega \right)$ as function of frequency. The inset shows the asymptotic $\sim 1/\omega$ behavior in the large-$\omega$ limit in the Kondo regime for $\langle n\rangle =0.985$. The temperature was fixed to $T={10}^{-5}\Gamma$.

Figure 7

The temperature dependence of $L_{ij}^{*}$ in the Kondo regime. The results are obtained by using the sum rule expression (B6).

Figure 8

(Color online) The temperature dependence of $\mathrm{Re}{L}_{ij}^{\left(0\right)}$. Here, $L_{11}^{\left(0\right)}$ can be identified with the dc conductance itself.