Renormalization group transport theory for open quantum systems: Charge fluctuations in multilevel quantum dots in and out of equilibrium

We present the real-time renormalization group (RTRG) method as a method to describe the stationary state current through generic multilevel quantum dots in nonequilibrium. The employed approach consists of a very rudimentary approximation for the renormalization group (RG) equations which neglects all vertex corrections while it provides a means to compute the effective dot Liouvillian self-consistently. Being based on a weak-coupling expansion in the tunneling between dot and reservoirs, the RTRG approach turns out to reliably describe charge fluctuations in and out of equilibrium for arbitrary coupling strength, even at zero temperature. We confirm this in the linear response regime with a benchmark against highly accurate numerical renormalization group data in the exemplary case of three-level quantum dots. For small to intermediate bias voltages and weak Coulomb interactions, we find an excellent agreement between RTRG and functional renormalization group data, which can be expected to be accurate in this regime. As a consequence, we advertise the presented RTRG approach as an efficient and versatile tool to describe charge fluctuations in quantum dot systems.

Figure 1

Linear conductance $G$ as function of the gate voltage for the model with ${\underline{\underline{t}}}^{\alpha }$ defined by Table 1, and level spacings $h_l/\mathrm{\Gamma }=(-0.7,0.0,0.5)$. We set $D=1000.0\mathrm{\Gamma }$ for all numerical RTRG calculations in this paper. All three applied methods (NRG, fRG, and RTRG) are in agreement regarding the position and shape of the conductance peaks.

Figure 2

Linear conductance $G$ as function of the gate voltage $V_{\text{g}}$ for the model with ${\underline{\underline{t}}}^{\alpha }$ defined by Table 1, $U=20.0\mathrm{\Gamma }$ and level spacings $h_l/\mathrm{\Gamma }=(-0.7,0.0,0.5)$. The inset shows a closeup of the central peak, clearly revealing the deviations in position and shape of the maximum within the fRG solution. In contrast, NRG and RTRG data are in good agreement regarding the position and the width of the conductance peak.

Figure 3

Conductance $G$ as function of the gate voltage $V_{\text{g}}$ for a model with tunneling matrix ${\underline{\underline{t}}}^{\alpha }$ defined in Table 2 and $h_l/\mathrm{\Gamma }=(-0.8,0.0,1.1)$ for small to intermediate Coulomb interactions, i.e., $U=0.1\mathrm{\Gamma }$ (left panel) and $U=1.0\mathrm{\Gamma }$ (right panel).

Figure 4

Conductance $G$ as function of the gate voltage $V_{\mathrm{g}}$ for a model with tunneling matrix ${\underline{\underline{t}}}^{\alpha }$ defined in Table 2, $h_l/\mathrm{\Gamma }=(-0.8,0.0,1.1)$ and $V=5.0\mathrm{\Gamma }$. While there is a very good agreement between fRG and RTRG solution for small Coulomb interactions $U=0.1\mathrm{\Gamma }$, the results from both approaches coincide only for the outer, i.e., the very left and the very right, peaks for moderate interaction strengths $U=2.0\mathrm{\Gamma }$. In the latter case, the solutions differ significantly in the region between the outer peaks, as explained in the main text.

Figure 5

RTRG solution for the conductance $G$ as function of the gate voltage $V_{\text{g}}$ for the model with ${\underline{\underline{t}}}^{\alpha }$ defined by Table 2, $U=20.0\mathrm{\Gamma }$, level spacings $h_l/\mathrm{\Gamma }=(-0.8,0.0,1.1)$, and different values for the bias $V$. Each of the three peaks occurring in the linear response regime ($V=0$) splits into two peaks of reduced height for increasing bias voltage $V=\mathrm{\Gamma }$. In contrast, additional resonance lines emerge for large enough bias ($V=5.0\mathrm{\Gamma }$).