Kondo model in nonequilibrium: Interplay between voltage, temperature, and crossover from weak to strong coupling

We consider an open quantum system in contact with fermionic metallic reservoirs in a nonequilibrium setup. For the case of spin, orbital, or potential fluctuations, we present a systematic formulation of real-time renormalization group at finite temperature, where the complex Fourier variable of an effective Liouvillian is used as flow parameter. We derive a universal set of differential equations free of divergencies written as a systematic power series in terms of the frequency-independent two-point vertex only, and solve it in different truncation orders by using a universal set of boundary conditions. We apply the formalism to the description of the weak to strong coupling crossover of the isotropic spin-$\frac{1}{2}$ nonequilibrium Kondo model at zero magnetic field. From the temperature and voltage dependence of the conductance in different energy regimes, we determine various characteristic low-energy scales and compare their universal ratio to known results. For a fixed finite bias voltage larger than the Kondo temperature, we find that the temperature dependence of the differential conductance exhibits nonmonotonic behavior in the form of a peak structure. We show that the peak position and peak width scale linearly with the applied voltage over many orders of magnitude in units of the Kondo temperature. Finally, we compare our calculations with recent experiments.

Figure 1

The analytic structure of the resolvent $1/[E-L(E\left)\right]$ for the isotropic Kondo model at zero temperature and finite voltage $V$. There are two pole positions at $E=0$ and $E=-i{\Gamma }^{*}$ together with branch cuts of $L\left(E\right)$ starting at $E=-i{\Gamma }^{*}+nV$ with some integer $n$.

Figure 2

The differential conductance $G(T,V)$ as function of temperature and voltage, scaled by the unitary conductance $G_0=G(T=V=0)=2e^2/h$. All RG equations were considered in third-order truncation and solved numerically.

Figure 3

The differential conductance $G_V\left(T\right)$ as function of temperature for constant voltage. The solid (dashed) lines show the results of the calculations in third- (second-) order truncation.

Figure 4

The position $T_{\text{max}}$ of the peak in the differential conductance $G_V\left(T\right)$, calculated within the third-order truncation scheme, at constant voltage $V$ (cf. the solid lines in Fig. 3). The solid line corresponds to the linear fit $\frac{T}{T_K^{*}}=0.597653\frac{V}{T_K^{**}}$ at large voltages.

Figure 5

The position $T_{\text{max}}$ of the peak in the differential conductance $G_V\left(T\right)$, calculated within the second-order truncation scheme, at constant voltage $V$ (cf. the dashed lines in Fig. 3). The solid line corresponds to the linear fit $\frac{T}{T_K^{*}}=0.399717\frac{V}{T_K^{**}}$ at large voltages.

Figure 6

The width $T_2\left(V\right)-T_1\left(V\right)$ of the peak in the differential conductance $G_V\left(T\right)$ at finite voltage $V$ (cf. Fig. 3). $T_1\left(V\right)$ and $T_2\left(V\right)$ are the temperatures for which $G_V\left[T_i\left(V\right)\right]-G_V(T=0)=\frac{1}{2}[G_V(T=0)+G_V\left(T_{\text{max}}\left(V\right)\right]$, $i=1,2$. The solid line corresponds to the linear fit $\frac{T}{T_K^{*}}=0.597653\frac{V}{T_K^{**}}$ that has been used in Fig. 4.

Figure 7

Quadratic fit for the differential conductance $G\left(T\right)\equiv G_{V=0}\left(T\right)$ in second-order truncation at low temperatures and $G\left(V\right)\equiv G_{T=0}\left(V\right)$ at low voltages, respectively. The differential conductance is plotted in such a way that the coefficients $c_T^{*}$ and $c_V^{*}$ in the expansions $G\left(T\right)/G_0\approx 1-c_T^{*}{\left(\frac{T}{T_K^{*}}\right)}^2$ and $G\left(V\right)/G_0\approx 1-c_V^{*}{\left(\frac{V}{T_K^{*}}\right)}^2$ can be identified easily.

Figure 8

Quadratic fit for the differential conductance $G\left(T\right)\equiv G_{V=0}\left(T\right)$ in third-order truncation at low temperatures and $G\left(V\right)\equiv G_{T=0}\left(V\right)$ at low voltages, respectively (cf. Fig. 7).

Figure 9

Comparison of the differential conductance $G_T\left(V\right)$ for fixed $T$ calculated with RTRG (solid lines) with experimental data from Ref. [37] (dots).

Figure 10

Comparison of the differential conductance $G_V\left(T\right)$ for fixed $V$ calculated with RTRG (solid lines) with experimental data from Ref. [37] (dots).