Hierarchical equations of motion approach to transport through an Anderson impurity coupled to interacting Luttinger liquid leads

We generalize the hierarchical equations of motion method to study electron transport through a quantum dot or molecule coupled to one-dimensional interacting leads that can be described as Luttinger liquids. Such leads can be realized, for example, by quantum wires or fractional quantum Hall edge states. In comparison to noninteracting metallic leads, Luttinger liquid leads involve many-body correlations and the single-particle tunneling density of states shows a power-law singularity at the chemical potential. Using the generalized hierarchical equations of motion method, we assess the importance of the singularity and the next-to-leading order many-body correlations. To this end, we compare numerically converged results with second- and first-order results of the hybridization expansion that is inherent to our method. As a test case, we study transport through a single-level quantum dot or molecule that can be described by an Anderson impurity model. Cotunneling effects turn out to be most pronounced for attractive interactions in the leads or repulsive ones if an excitonic coupling between the dot and the leads is realized. We also find that an interaction-induced negative differential conductance near the Coulomb blockade thresholds is slightly suppressed as compared to a first-order and/or rate equation result. Moreover, we find that the two-particle ($n$-particle) correlations enter as a second-order ($n$-order) effect and are, thus, not very pronounced at the high temperatures and parameters that we consider.

Figure 1

Graphical scheme of the transport setup that we consider, i.e., a quantum dot coupled to two, semi-infinite Luttinger liquid leads with chemical potentials that differ by the applied bias voltage.

Figure 2

Single-particle tunneling density of states, $J\left(\omega \right)$, at the edge of a Luttinger liquid lead for attractive $\left(Y=0.8\right)$ and repulsive interactions $\left(Y=1.2\right)$ with $T=200$ K and $W=4$ eV. $\mathrm{\Gamma }$ is the system-lead coupling strength.

Figure 3

Current-voltage characteristics for the quantum dot at the charge-symmetric point $\left(\epsilon =-U/2\right)$, for various interaction parameters of the Luttinger liquid leads ($Y=0.8/1/1.2$ in the upper/middle/lower panels) and temperatures ($T=100/200$ K in the left/right panels). Solid (dashed) lines are obtained by HQME (rate equations).

Figure 4

Current-voltage characteristics for the quantum dot far away from the charge-symmetric point $\left(\epsilon =U/2\right)$, for various interaction parameters of the Luttinger liquid leads ($Y=0.8/1/1.2$ in the upper/middle/lower panels) and temperatures ($T=100/200$ K in the left/right panels). Solid (dashed) lines are obtained by HQME (rate equations).

Figure 5

Comparison of current-voltage characteristics for the quantum dot at the charge-symmetric point $\left(\epsilon =-U/2\right)$ for $Y=0.8$ and $T=100$ K obtained with rate equations (blue line), full first-order (dashed green line), second-order (orange line), and the converged HQME results (dashed purple line).

Figure 6

The top panel shows the current-voltage characteristics with and without contributions of two-particle correlations $C^{\left(2\right)}$, i.e., $\widehat{\rho }$ ADO's. The bottom panel shows the dot populations, Eq. (34), with and without contributions of two-particle correlations $C^{\left(2\right)}$.

Figure 7

$|{\widehat{C}}^{\left(2\right)}(0,0,t_1,t_2)|$ at $T=200$ K for $Y=1.2$. (a) $W=4$ and (b) 20 eV. The correlation function decays exponentially.