Floquet renormalization group approach to the periodically driven Kondo model

We study the interplay of strong correlations and coherent driving by considering the strong coupling Kondo model driven by a time-periodic bias voltage. Combining a recent nonequilibrium renormalization group method with Floquet theory, we find that by the coherent dressing of the driving field side replicas of the Kondo resonance emerge in the conductance, which are not completely washed out by the decoherence induced by the driving. We show that to accurately capture the interplay of driving and strong correlations one needs to go beyond simple phenomenological pictures, which underestimate decoherence, or adiabatic approximations, highlighting the relevance of non-Markovian memory effects. Within our method the differential conductance shows good quantitative agreement with experimental data in the full crossover regime from weak to strong driving. We analyze memory effects in detail based on the response to short voltage pulses.

Figure 1

Sketch of the model. A periodic bias voltage is applied across a quantum dot, which in the Kondo limit can be reduced to a single spin-$\frac{1}{2}$ (red) coupled to two fermionic reservoirs $L$ and $R$ (blue).

Figure 2

Differential conductance $G_{\mathrm{avg}}=d{I}_{\mathrm{avg}}/d{V}_{\mathrm{avg}}$ and $G_{\mathrm{osc}}=d{I}_{\mathrm{osc}}/d{V}_{\mathrm{osc}}$ for $V\left(t\right)=V_{\mathrm{avg}}+V_{\mathrm{osc}}cos\left(\mathrm{\Omega }t\right)$ as a function of $V_{\mathrm{avg}}$ and $V_{\mathrm{osc}}$ at $\mathrm{\Omega }=7.55T_K$. Red lines indicate the parameters which are compared to approximations in Figs. 3 and 3. The time-averaged differential conductance (upper panel) shows peaks at $V_{\mathrm{avg}}=n\mathrm{\Omega }$ weighted by $J_n{(V_{\mathrm{osc}}/\mathrm{\Omega })}^2$. The width of these peaks in $G_{\mathrm{avg}}\left(V_{\mathrm{avg}}\right)$ at constant $V_{\mathrm{osc}}$ is of the order of $T_K$ while all other structures are smooth on the energy scale of $\mathrm{\Omega }$. $G_{\mathrm{osc}}$ at constant $V_{\mathrm{avg}}$ or $V_{\mathrm{osc}}$ shows a broad maximum at $V_{\mathrm{avg}}\approx V_{\mathrm{osc}}$. This result can be understood from the low-frequency limit, where $V_{\mathrm{avg}}=V_{\mathrm{osc}}$ implies that $V\left(t\right)\approx 0$ for a relatively long time in each period, which leads to a high differential conductance due to the Kondo resonance.

Figure 3

Differential conductance $G_{\mathrm{avg}}(V_{\mathrm{avg}},V_{\mathrm{osc}},\mathrm{\Omega })$ from FRTRG (red line) compared to the adiabatic approximation (black dotted line), the phenomenological Eq. (160) (thin blue line), and the analytic approximation for limiting cases from Ref. [16] [dashed lines in panels (c) and (d)]. We use $\mathrm{\Omega }=7.55T_K$ as in Fig. 2 except in panel (c). Many features of these curves are due to the replicas of the Kondo resonance and are overestimated by the phenomenological approximation. Oscillations of $G_{\mathrm{avg}}$ as a function of $\mathrm{\Omega }$ and $V_{\mathrm{osc}}$ at intermediate frequencies and high $V_{\mathrm{osc}}$ are not captured at all by the analytic approximations of Ref. [16]. As expected, the adiabatic approximation (defined by setting $\mathrm{\Omega }\to 0$) is only applicable if $\mathrm{\Omega }\ll max\{V_{\mathrm{osc}},V_{\mathrm{avg}},T_K\}$.

Figure 4

Comparison of FRTRG data (red and cyan lines) to measured differential conductance $G_{\mathrm{avg}}=d{I}_{\mathrm{avg}}/d{V}_{\mathrm{avg}}$ through a single electron transistor in the Kondo regime from Ref. [32] (black dots) at $\mathrm{\Omega }/2\pi =13.47\mathrm{GHz}$ and $T_K\approx 0.5\mathrm{\Omega }$. The red curves are calculated for symmetric coupling to the reservoirs ($x_L=x_R$) and rescaled by a factor 0.114 to include finite temperature and asymmetry effects, whereas the cyan lines are obtained for an asymmetry of $x_L=0.05$ in the coupling and are only rescaled by a factor 0.62 to account for finite temperature. The average voltage $V_{\mathrm{avg}}$ in the theory curves is shifted to match the position of the central Kondo peak in the experimental data. The FRTRG data for asymmetric coupling show more pronounced peaks which are, however, expected to be smeared out by the finite temperature. The curves show $\mathrm{\Omega }=2.11T_K$ and (starting from the top) $V_{\mathrm{osc}}=0.73\mathrm{\Omega },1.13\mathrm{\Omega },1.51\mathrm{\Omega },1.68\mathrm{\Omega }$, and $3.62\mathrm{\Omega }$. We adapted the calibration of $V_{\mathrm{osc}}$ because the procedure of calibrating $V_{\mathrm{osc}}$ in Ref. [32] is expected to be biased (see Appendix pp8). The scale of the $y$ axis corresponds to the curve at the bottom and subsequent curves are separated by $0.04e^2/h$ for clarity.

Figure 5

Current in time and Fourier domain for $V\left(t\right)=V_{\mathrm{osc}}cos\left(\mathrm{\Omega }t\right), \mathrm{\Omega }=7.55T_K$. (a) Higher harmonics in the response current relative to the first harmonic. Even modes vanish at $V_{\mathrm{avg}}=0$ due to symmetry. The decay is approximately algebraic for $n<V_{\mathrm{osc}}/\mathrm{\Omega }$ in agreement with the adiabatic limit $\mathrm{\Omega }\to 0$ (black), except for very high harmonics ($n\gtrsim 40$ for $V_{\mathrm{osc}}=80\mathrm{\Omega }$). For $n>V_{\mathrm{osc}}/\mathrm{\Omega }$ the modes decay exponentially. (b) Time-resolved current for $V_{\mathrm{osc}}=2.5\mathrm{\Omega },5\mathrm{\Omega },10\mathrm{\Omega }$, and $20\mathrm{\Omega }$ (from red to green). Markers indicate times $t$ at which ${\int }_{t^{*}}^{t}V\left(s\right)ds$ is an integer multiple of $2\pi$. The reference time $t^{*}=-T/4$ is chosen such that $V\left(t^{*}\right)=0$. (c) $I\left(t\right)$ vs ${\int }_{t^{*}}^{t}V\left(s\right)ds$ parametrized by $t$ for $V_{\mathrm{osc}}=2.5\mathrm{\Omega }...20\mathrm{\Omega }$ in steps of $2.5\mathrm{\Omega }$. The coherent oscillations in this representation reveal the origin of the oscillations in $I\left(t\right)$ as explained in the text.

Figure 6

Current and differential conductance as a function of time for a Gaussian voltage pulse centered at $t=0$ of duration (full width at half maximum) $t_{\mathrm{pulse}}=0.058/T_K$. For $\phi =2\pi$ or $\pi$ the current quickly decays to zero after the pulse, while for $\phi =\pi /2$ or $3\pi /2$ a slower decaying current persists after the pulse. For $\phi \approx 3\pi /2$ this current after the pulse is negative. The axes are chosen such that the areas under the curves for $I\left(t\right)$ can be directly compared between the left and the right panel. The differential conductance $G\left(t\right)=dI\left(t\right)/d{V}_{\mathrm{avg}}$ indicates the recovery of the Kondo resonance, which happens on a time scale of the order of $1/T_K$, slightly slower than the decay of the current.

Figure 7

Relevance of the phase difference $\phi =\int dtV\left(t\right)$. (a) Average charge $Q$ transported by a single Gaussian voltage pulse of constant duration $t_{\mathrm{pulse}}=0.33/T_K$ and variable height. (b) Charge $Q$ for short pulses of constant phase $\phi =\pi /2,\pi ,3\pi /2$, and $2\pi$ (blue, green, orange, and magenta). $t_{\mathrm{pulse}}$ is the full width at half maximum of a Gaussian pulse. (c), (d) Time $t_{\mathrm{recover}}$ needed to recover the Kondo resonance after a Gaussian voltage pulse for the same $V\left(t\right)$ as in panels (a) and (b). The effect of $\phi$ on the recovery time clearly increases for shorter pulses.

Figure 8

Relative difference $(G_{\mathrm{avg}}-G_{\mathrm{avg}}^{\text{leading}\text{order}})/G_{\mathrm{avg}}$ of the differential conductance for $\mathrm{\Omega }=7.55T_K$. Here $G_{\mathrm{avg}}^{\text{leading}\text{order}}$ is the differential conductance from an FRTRG calculation that keeps only the leading order in $J$. For all other data we include the next-to-leading order in $J$. The dark red spots are the peaks in the $G$ which are overestimated in their height by $G_{\mathrm{avg}}^{\text{leading}\text{order}}$. The maximum deviation for these parameters is approximately $10\%$ in the regime where the energy scales $V_{\mathrm{osc}}$ and $\mathrm{\Omega }$ are approximately equal.

Figure 9

Relative difference $(G_{\mathrm{avg}}^{\delta {\mathrm{\Gamma }}^{\gamma }}-G_{\mathrm{avg}}^{{\mathrm{\Gamma }}^{\gamma }})/G_{\mathrm{avg}}^{\delta {\mathrm{\Gamma }}^{\gamma }}$ of the differential conductance $G_{\mathrm{avg}}^{\delta {\mathrm{\Gamma }}^{\gamma }}=\delta {\mathrm{\Gamma }}^{\gamma }{\left(0\right)}_{00}$ and the numerical derivative of the current $G_{\mathrm{avg}}^{{\mathrm{\Gamma }}^{\gamma }}=d{\mathrm{\Gamma }}^{\gamma }{\left(0\right)}_{00}/d{V}_{\mathrm{avg}}$ for $\mathrm{\Omega }=7.55T_K$. The maximum deviation for these parameters is approximately $3.5\%$ and $G_{\mathrm{avg}}^{{\mathrm{\Gamma }}^{\gamma }}$ is in general slightly larger than $G_{\mathrm{avg}}^{\delta {\mathrm{\Gamma }}^{\gamma }}$ except near equilibrium. The values for $G_{\mathrm{avg}}^{\delta {\mathrm{\Gamma }}^{\gamma }}$ used in this comparison are shown in Fig. 2.

Figure 10

Differential conductance $G_{\mathrm{avg}}$ in units of $2e^2/h$ as calculated using FRTRG for $\mathrm{\Omega }=0.75T_K$ (upper panel) and its adiabatic approximation (lower panel). The comparison illustrates that in the full FRTRG results the maximum in $G_{\mathrm{avg}}$ is shifted slightly towards smaller $V_{\mathrm{avg}}$. Thus a calibration of $V_{\mathrm{osc}}$ by comparison to the adiabatic limit will be biased and underestimate $V_{\mathrm{osc}}$. We stress that this does not simulate the precise Kondo temperature used for the calibration in Ref. [32], but only aims for a qualitative comparison with the calibration measurement shown in Fig. 2 of Ref. [32].