Width of the charge-transfer peak in the SU($N$) impurity Anderson model and its relevance to nonequilibrium transport

We calculate the width $2{\mathrm{\Delta }}_{\text{CT}}$ and intensity of the charge-transfer peak (the one lying at the on-site energy $E_d$) in the impurity spectral density of states as a function of $E_d$ in the SU($N$) impurity Anderson model (IAM). We use the dynamical density-matrix renormalization group (DDMRG) and the noncrossing approximation (NCA) for $N=4$ and a 1/$N$ variational approximation in the general case. In particular, while for $E_d\gg \mathrm{\Delta }$, where $\mathrm{\Delta }$ is the resonant level half-width, ${\mathrm{\Delta }}_{\text{CT}}=\mathrm{\Delta }$ as expected in the noninteracting case, for $-E_d\gg N\mathrm{\Delta }$ one has ${\mathrm{\Delta }}_{\text{CT}}=N\mathrm{\Delta }$. In the $N=2$ case, some effects of the variation of ${\mathrm{\Delta }}_{\text{CT}}$ with $E_d$ were observed in the conductance through a quantum dot connected asymmetrically to conducting leads at finite bias [J. Könemann et al., Phys. Rev. B 73, 033313 (2006)]. More dramatic effects are expected in similar experiments that can be carried out in systems of two quantum dots, carbon nanotubes or other, realizing the SU(4) IAM.

Figure 1

DDMRG correction vector data of the CT at $\eta =10\mathrm{\Delta }$. Open circles (crosses) for $N_s=40$ ($N_s=80$), for $E_d=40\mathrm{\Delta }$ (see Fig. 3). In legends we indicate the ${\eta }_{\text{eff}}$ obtained from a fit with a Lorentzian $\propto K/({(\omega -{\omega }_{up})}^2+{\eta }_{\text{eff}}^2)$ (thin line) for each size.

Figure 2

Impurity spectral density calculated with the NCA (full lines) and variational wave function (dashed lines) for $E_d=-15\mathrm{\Delta }$.

Figure 3

Half width at half maximum of the charge transfer peak as a function of the effective on site energy using NCA (green circles) and DDMRG (blue squares). Dashed line is a fit of the filled symbols using Eq. (18).

Figure 4

Shift of the position of the CT peak as a function of on-site energy calculated with NCA for different temperatures $T$.

Figure 5

Half width at half maximum of the charge transfer peak as a function of the effective on site energy using NCA at two different temperatures $T$. Dashed lines are fits using Eq. (18).

Figure 6

Total occupancy as a function of on-site energy level calculated with NCA for the SU($N$) IAM with $N=2$ and 4. Dashed blue line corresponds to the Bethe ansatz (BA) result for $N=2$ (see text). Dotted lines are the noninteracting results.

Figure 7

Scheme of the Fermi level of both leads and the displacement of the dot level $E-d$ due to capacitance effects. The top (bottom) figures represent different situations for positive (negative) $E_d^0$. When the Fermi level of one of the leads coincides with $E_d$ a peak in the conductance is observed. Assuming that the right lead is more strongly coupled to the dot, the system can be in Kondo, intermediate valence (IV), or empty orbital regime (EOR) as indicated.

Figure 8

Weight of the CT peak obtained from the fit of the NCA CT peak (circles and squares) and from Eqs. (23) and (24) (dashed lines) with $N=4$ as a function of the effective impurity level for several temperatures $T$. Blue dashed line corresponds to $T=0$.

Figure 9

Differential conductance for an SU(4) impurity as a function of bias voltage for asymmetric couplings ${\mathrm{\Delta }}_R\gg {\mathrm{\Delta }}_L$, capacitance ratio $\alpha =1/2$, $E_d^0=15\mathrm{\Delta }$ and several temperatures $T$. The constant $C_1^{\prime }=eC/\left(2\pi \mathrm{\Delta }\right)$.

Figure 10

Same as Fig. 9 for $E_d^0=-15\mathrm{\Delta }$. The constant $C_2^{\prime }=eC/\left(32\pi \mathrm{\Delta }\right)$.