Renormalization group analysis of the interacting resonant-level model at finite bias: Generic analytic study of static properties and quench dynamics

Using a real-time renormalization group method we study the minimal model of a quantum dot dominated by charge fluctuations, the two-lead interacting resonant level model, at finite bias voltage. We develop a set of RG equations to treat the case of weak and strong charge fluctuations, together with the determination of power-law exponents up to second order in the Coulomb interaction. We derive analytic expressions for the charge susceptibility, the steady-state current, and the conductance in the situation of arbitrary system parameters, in particular away from the particle-hole symmetric point and for asymmetric Coulomb interactions. In the asymmetric situation we find that power laws can be observed for the current only as a function of the level position (gate voltage) but in general not as a function of the voltage except for extremely large voltages. Furthermore, we study the quench dynamics after sudden switch-on of the level-lead couplings. The time evolution of the dot occupation and current is governed by exponential relaxation accompanied by voltage-dependent oscillations and characteristic algebraic decay.

Figure 1

Sketch of the interacting resonant level model. A local level with energy ${\epsilon }_0$ is coupled via hoppings $t_{L/R}^{(0)}$ and Coulomb couplings $U_{L/R}^{(0)}$ to two spinless fermionic reservoirs held at chemical potentials ${\mu }_{L/R}=\pm V/2$.

Figure 2

Results for the static susceptibility $\chi$ for the symmetric model with $U_L^{(0)}=U_R^{(0)}=U^{(0)}$ and $\Gamma {}_L^{(0)}=\Gamma {}_R^{(0)}=\Gamma {}^{(0)}$ at ${\epsilon }_0=V=0$. Upper panel: comparison of RTRG-FS (solid lines) results, the analytic solution of the flow equations (dashed lines), and NRG data (symbols). Lower panel: exponent $\beta =-g/(1+g)$.

Figure 3

Results for the current $I(V)$ for asymmetric Coulomb interactions $U_{L/R}^{(0)}=(1\pm {\gamma }_U)0.1/\pi$ with ${\gamma }_U=0.75$, 0.5, 0.25, 0 from top to bottom and $t_L^{(0)}=t_R^{(0)}=0.001$, ${\epsilon }_0=0$; the numerical solution (solid lines) is compared to the analytical result (dashed lines). Lower panel: logarithmic derivative.

Figure 4

Results for the current $I(V)$ for the symmetric model with $t_L^{(0)}=t_R^{(0)}=0.001$ and $U_L^{(0)}=U_R^{(0)}=0.1/\pi$ of resonance (solid lines) and on resonance (dashed lines). Inset: logarithmic derivative.

Figure 5

Conductance $G(V)=\mathit{dI}/\mathit{dV}$ for the symmetric model with $t_L^{(0)}=t_R^{(0)}=0.001$, $U_L^{(0)}=U_R^{(0)}=0.1/\pi$ and different values of the gate voltage ${\epsilon }_0=0$, 0.1, 1, 2, 5, $10\hspace{0.5em}{T}_K$ from top to bottom.

Figure 6

Results for the conductance $G({\epsilon }_0)$ for asymmetric Coulomb interactions $U_{L/R}^{(0)}=(1\pm {\gamma }_U)0.1/\pi$ with ${\gamma }_U=0$, 0.25, 0.5, 0.75 from top to bottom and $t_L^{(0)}=t_R^{(0)}=0.001$ at $V=0$; the obtained power law is characterized by the exponent $~$$2(g+1)$ in linear order, with $g=2U^{(0)}$.

Figure 7

Time evolution of the dot occupation $n(t)$ for $V={\epsilon }_0=10\hspace{0.5em}{T}_K$ and different values of $U^{(0)}$. The initial condition is given by $n(0)=0$. We observe oscillating behavior at short times, $T_{K}t⩽5$.

Figure 8

Time evolution of the dot occupation $n(t)$ for $U^{(0)}=0.1/\pi$, $V=100\hspace{0.5em}{T}_K$, and different values of ${\epsilon }_0$. The initial condition is given by $n(0)=0$.

Figure 9

Time evolution of the current $I_L(t)$ in the left lead for $U^{(0)}=0.1/\pi$, ${\epsilon }_0=10\hspace{0.5em}{T}_K$, and different values of $V$. The initial condition is given by $n(0)=0$. The nonzero current at $t=0$ is determined by the displacement current $\mathit{dn}(t)/\mathit{dt}$. The current oscillates with frequency $~{\epsilon }_0-V/2$; note the absence of oscillations on resonance.

Figure 10

Diagrams of the discrete RG step. Double vertices are represented by dots lying close to each other. The indices $s/a$ refer to the symmetric/antisymmetric part of the reservoir contraction.

Figure 11

Diagrams of the RG equations. Double vertices are represented by dots lying close to each other. For ${\mathrm{Ḡ}}_{{11}^{\prime }}$, diagrams have to be subtracted where the indices $1$ and $1^{\prime }$ are interchanged, provided this gives a new diagram. This guarantees the relation ${\mathrm{Ḡ}}_{{11}^{\prime }}=-{\mathrm{Ḡ}}_{1^{\prime }1}$.

Figure 12

Analytic structure of the integrand of $J_{\pm }(t)$. All singularities appear in the lower half plane. Red dots stand for poles while red solid lines represent branch cuts. The pole at $z=0$ corresponds to the stationary state. The blue dashed and green dotted lines are the original and deformed integration contours, respectively.

Figure 13

Integration contour for $J_{\pm }(t)$ on resonance $|\mathrm{\epsilon ̃}-V/2|\ll T_K$.