Functional renormalization group in Floquet space

We present an extension of the functional renormalization group to Floquet space, which enables us to treat the long time behavior of interacting time periodically driven quantum dots. It is one of its strength that the method is neither bound to small driving amplitudes nor to small driving frequencies, i.e., very general time periodic signals can be considered. It is applied to the interacting resonant level model, a prototype model of a spinless, fermionic quantum dot. The renormalization in several setups with different combinations of time periodic parameters is studied, where the numerical results are complemented by analytic expressions for the renormalization in the limit of small driving amplitude. We show how the driving frequency acts as an infrared cutoff of the underlying renormalization group flow which manifests in novel power laws. We utilize the tunability of the effective reservoir distribution function in a periodically driven onsite energy setup to show how its shape is directly reflected in the renormalization group flow. This allows us to flexibly tune the power-law renormalization generically encountered in quantum dot structures. Finally, an in-phase quantum pump as well as a single parameter pump are investigated in the whole regime of driving frequency, demonstrating that the new power law in the driving frequency is reflected in the mean current of the latter.

Figure 1

Diagrammatic representation of the first flow equation in the time dependent form. The hierarchy of flow equations is cut after this equation by setting the two-particle vertex function to the bare interaction [see Eq. (42)]. To deal with time periodic systems the flow equation is transformed to Floquet space.

Figure 2

Sketch of the interacting resonant level model, a prototype model for an interacting quantum dot. Within our method, any of the shown parameters can be considered as time periodic. The time dependency of ${\mu }_{\mathrm{L}\left(\mathrm{R}\right)}\left(t\right)$ can be shifted to the hopping amplitudes ${\tau }_{\mathrm{L}\left(\mathrm{R}\right)}$ in the scaling limit.

Figure 3

Numerical data (solid line) and analytic expressions (crosses) for the renormalization flows of the $k=0,1$ components for protocol 1 and 2 for $U/D=0.1$ and $T_{\mathrm{K}}/D=4.62\times {10}^{-5}$ for several values of driving frequency $\mathrm{\Omega }$. The arrows in the upper row indicate the position of the respective driving frequencies. The upper panel shows the $k=0$ component of the left hopping ${\tau }_{\mathrm{L}}$ for all protocols for different values of driving frequency. There is no dependence on the driving frequency $\mathrm{\Omega }$. In contrast, the middle panel shows the $k=1$ component, which clearly exhibits a dependence on the driving frequency. $\mathrm{\Omega }$ provides the cutoff of the flow if $\mathrm{\Omega }>T_{\mathrm{K}}$ in protocol 2 if ${\tau }_{\mathrm{L}}$ is time periodic. The third panel shows the flow of ${\tau }_{\text{L},k=1}$ if both ${\tau }_{\mathrm{L}}$ and ${\tau }_{\mathrm{R}}$ are driven. To the order $\mathcal{O}\left(p\right)$ the flows equal the ones of protocol 1, because the feedback of ${\tau }_{\mathrm{R}}$ is of higher order.

Figure 4

Numerical data (sold line) and analytic expressions (symbols) for the renormalized left hopping and the onsite energy as a function of driving frequency $\mathrm{\Omega }$ for protocols 1, 3, and 4 for $U/D=0.1$ and $T_{\mathrm{K}}/D=4.62\times {10}^{-5}$. The absolute values of the first five higher harmonics are shown in the first panel as a function of $k\mathrm{\Omega }$. The left hopping is driven with ${\tau }_{\mathrm{L}}={\tau }_0+{\sum }_{k=1}^{10}\mathrm{\Delta }\tau sin\left(k\mathrm{\Omega }t\right)$, i.e., each harmonic has the same initial value. All solid lines lie on top of each other demonstrating the same functional dependence on $k\mathrm{\Omega }$ for each $k\mathrm{th}$ harmonic. When $k\mathrm{\Omega }\le T_{\mathrm{K}}$, (shown in the inset), ${\tau }_{\text{L},k=1}$ bends to an $\mathrm{\Omega }$ independent value, confirming that $T_{\mathrm{K}}$ is the main energy scale in the adiabatic limit. The middle panel shows the renormalization of ${\tau }_{\text{L},k=1}$ as well as ${\epsilon }_{k=1}$ for oscillating onsite energy. While ${\tau }_{\text{L},k=1}$ is renormalized strongly, ${\epsilon }_{k=1}$ is not and its feedback into ${\tau }_{k=1}^{\mathrm{ren}}$ is negligible. The third panel shows the renormalization of ${\tau }_{L,k=1}$ in protocol 4. Here the ${\epsilon }_{k=1}$ is chosen five times larger compared to the other setups, to make its contribution in the form of the bump more visible.

Figure 5

The exponents of the renormalization of the zeroth and first harmonic of the hopping in different regimes as well as the corresponding exponent of the mean current $J_0$. All exponents are extracted from the logarithmic derivative (implemented as centered differences) of the full numerical solution. For small $U$ the exponents show excellent agreement with the analytic predictions. The exponents ${\alpha }_{k=0/1}$ show stronger higher order corrections in $U$ than the exponents ${\alpha }_{\mathrm{\Omega }/J}$.

Figure 6

Numerical data (solid lines) and analytic expressions (symbols) of the renormalization flows of ${\tau }_{k=1}$ for protocol 3 and 4 for $U/D=0.1$ and $T_{\mathrm{K}}/D=4.62\times {10}^{-5}$ for several values of the driving frequency $\mathrm{\Omega }$. The arrows on the upper line indicate the respective value of $\mathrm{\Omega }$. The upper panel shows the renormalization group flow of the $k=1$ component of protocol 3, which is more complicated than the renormalization of the $k=1$ component of the protocols 1 and 2. While in the antiadiabatic limit the renormalization is suppressed by the driving frequency $\mathrm{\Omega }$, the interplay between $\mathrm{\Omega }$ and $\mathrm{\Lambda }$ at around $T_{\mathrm{K}}$ results in strong renormalization for $\mathrm{\Omega }\approx T_{\mathrm{K}}$ (compare also Fig. 4). The lower panel shows the renormalization of ${\tau }_{k=1}$ in protocol 4. The renormalization for the large values of driving frequency $\mathrm{\Omega }$ is only defined by the contribution of the driven ${\tau }_{\mathrm{L}}$, in the limit of adiabatic driving the flow shows deviations resulting from the contribution by the renormalization due to the nonzero ${\epsilon }_{k=1}$.

Figure 7

The frequency $\mathrm{\Omega }$ can act as cutoff even in the zeroth component: Driving only the onsite energy ${\epsilon }_0$ results in multistep effective reservoir distribution functions, with steps at $\omega =n\mathrm{\Omega }$ of width $\mathrm{\Omega }$ and height $h_n={\left|J_n\left(\frac{\mathrm{\Delta }\epsilon }{\mathrm{\Omega }}\right)\right|}^2$. The distribution functions are sketched at the right-hand side to picture the different situations. The effective distribution function is then reflected in the renormalization flow of the $k=0$ harmonic of the hopping. The renormalization (solid line) in protocol (a) shows a clear infrared cutoff due to the edge at $\omega =\mathrm{\Omega }$. The renormalizations (dashed line) in protocol (b) shows bending at two energy scales due to the edges at $w=0,\pm 2\mathrm{\Omega }$. The renormalization (dotted line) in protocol (c) is characterized by the multiple energy scale off the equidistant steps with equal height, resulting in a long tail. In the lower panel, the susceptibility of protocol (a) is compared to the susceptibility of a setup with an applied DC bias voltage. Due to similar renormalization of the zeroth component of the hopping, the susceptibility shows equal behavior for both situations. For the whole plot $U/D=0.2$ with $T_{\mathrm{K}}/D=7.93\times {10}^{-5}$.

Figure 8

Pumped charge for various phase differences $\theta$ between the signals of left hopping and onsite energy with $\mathrm{\Delta }\tau /{\tau }_0=\mathrm{\Delta }\epsilon /T_{\mathrm{K}}=0.05$ for $U/D=0.0,0.2$ and accordingly $T_{\mathrm{K}}/D=2.5\times {10}^{-5},7.93\times {10}^{-5}$. A monotonously decreasing function for $\theta =\pi /2$ with increasing driving frequency becomes a nonmonotonous function with maximal pumping in the opposite direction as the phase difference vanishes. The behavior is independent of interaction. In the limit of fast driving no charge is pumped at all, independent of the phase difference between the signals.

Figure 9

Mean left current as a function of onsite energy $\epsilon$ for protocol 1 with the noninteracting $T_K=0.01$. Only the left hopping element is varied time periodically with a sine signal in the noninteracting system. Results obtained with the quantum master equations are compared to the Landauer-Büttiker type formula extended to finite temperature. The quantum master equation is capable of reproducing the finite mean current for large driving frequency $\mathrm{\Omega }/\left(2\pi T_{\mathrm{K}}\right)=200$. With increasing temperature the agreement with the analytic formula improves.

Figure 10

The time periodically driven 1D system can be understood as a 2D system with replicas shifted by $\mathrm{\Omega }$. Neighboring replicas are coupled via ${\epsilon }_{k=1}$ for the setup with sinusoidally driven onsite energy. For the calculation of the renormalization of ${\tau }_{k=1}$ only the zeroth and first replica are of relevance to the order $\mathcal{O}\left(p\right)$. The green, dashed line indicates the coupling of interest, which is initially zero. To realize the compact effective model (as depicted on the right hand side), we take advantage of the left-right symmetry and include the first side into an effective reservoir in the $k=0$ channel. In this effective model ${\tau }_{\mathrm{L},\mathrm{k}=1}^{\mathrm{ren}}$ is the hopping between the first and third site of the central region as indicated.