Theory of a resonant level coupled to several conduction-electron channels in equilibrium and out of equilibrium

The spinless resonant level model is studied when it is coupled by hopping to one of the arbitrary numbers of conduction-electron channels. The Coulomb interaction acts between the electron on the impurity and in the different channels. In the case of a repulsive or attractive interaction the conduction electrons are pushed away or attracted to ease or hinder the hopping by creating unoccupied or occupied states, respectively. In the screening of the hopping orthogonality catastrophe plays an important role. At equilibrium in the weak- and strong-coupling limits the renormalizations are treated by perturbative, numerical, and Anderson-Yuval Coulomb gas methods. In the case of two leads the current due to applied voltage is treated in the weak-coupling limit. The presented detailed study should help to test other methods suggested for nonequilibrium transport.

Figure 1

Vertex diagrams (a), (b), (c) and the impurity self-energy (d). Solid lines stand for conduction-electron propagators while the dotted line for those of electron on the impurity level. The interactions are indicated by dots $\left(U\right)$ and crosses $\left(V\right)$. In the case of conduction electrons the channel indices are also indicated.

Figure 2

Vertex correction to the hybridization.

Figure 3

(Color online) The renormalized hybridization as a function of ${\varrho }_{0}U$ for different channel numbers. The intermediate maximum can diverge only for ${\varrho }_{0}U=1$; in all the other cases, the increases are also rather moderate.

Figure 4

(Color online) Upper panels: impurity density of states for $V∕D=0.015$ and $U∕D=0,...,0.3$ as obtained by perturbative RG and Wilson’s NRG. The lower panel shows the renormalized value of the hybridization, $V_{\mathrm{ren}}$, as a function of the interaction strength $U$. The numerical data supports the weak-coupling results for $U∕D\le 0.2$.

Figure 5

(Color online) Comparison between weak-coupling RG and NRG approaches: the renormalized value of the hybridization, $V_{\mathrm{ren}}$ as a function of the interaction strength $U$ for $N=2$ and $N=4$.

Figure 6

The solid lines represents the time interval when the impurity level is occupied, and the light ones stand for unoccupied levels. The conduction electrons can join the time line at the end points of the intervals where hopping $V_{\alpha }$ takes place while they can touch the time line at any points due to the Coulomb interaction. Those are indicated by dashed lines which are labeled according to the channel indices ${\alpha }_i$.

Figure 7

Panels (i)–(vi): the diagrams up to $\sim U^3$ order contributing to the numerator of Eq. (26). The diagrams should be decorated by the direction of the lines in all possible ways. The diagram of the self-energy is shown in panel (vii).

Figure 8

(Color online) Current obtained by weak coupling RG for $\Delta ∕D=0.1$ and $U∕D=0,0.005,0.03,0.05,0.10$. For this range of interaction strength the weak-coupling method was reliable in equilibrium.

Figure 9

(Color online) Current obtained by the scattering formalism for $\Delta ∕D=0.1$ and $U∕D=0,0.10,0.20,0.30,0.50,0.60,0.85$. Note that the weak-coupling result is reliable for $U∕D\le 0.2$ only.

Figure 10

(Color online) The two relevant diagrams up to $\sim U^3$ contributing to the numerator of Eq. (26). A Ward identity ensures the cancellation of these diagrams for small frequencies, meaning that no logarithmic term survives in second order and thus the invariant coupling $U_{\mathrm{inv}}=\mathrm{const}$.