Current noise of the interacting resonant level model

We study the zero-frequency current noise of the interacting resonant level model for arbitrary bias voltages using a functional renormalization group approach. For this we extend the existing nonequilibrium scheme by deriving and solving flow equations for the current-vertex functions. On-resonance artificial divergences of the latter found in lowest-order perturbation theory in the two-particle interaction are consistently removed. Away from resonance they are shifted to higher orders. This allows us to gain a comprehensive picture of the current noise in the scaling limit. At high bias voltages, the current noise exhibits a universal power-law decay, whose exponent is, to leading order in the interaction, identical to that of the current. The effective charge on resonance is analyzed in detail, employing properties of the vertex correction. We find that it is only modified to second or higher order in the two-particle interaction.

Figure 1

(a) Schematic picture of the IRLM. (b) The Keldysh contour.

Figure 2

Diagrammatic representation of the current noise. The circle represents the three-point vertex function. The left diagram is called the bubble term, while the right is called the vertex correction.

Figure 3

Diagrammatic representation of the flow equations of (a) the self-energy and (b) the three-point current-vertex function in the static approximation. The square represents the two-particle interaction.

Figure 4

The dependence of $\overline{I}$ on the dimensionless interaction $u$ for various $V$. The unit of energy, $T_{\mathrm{K}}$, is introduced in Sec. 3b. The parameters are $t/\mathrm{\Delta }=0.001$, $\epsilon /T_{\mathrm{K}}=0$, and $T/T_{\mathrm{K}}=0$.

Figure 5

The dependence of noise and its logarithmic derivative on $V$ for various $u$. The parameters are $\epsilon /T_{\mathrm{K}}=0$, $t/\mathrm{\Delta }=0.001$, and $T/T_{\mathrm{K}}=0$.

Figure 6

(a) The dependence of the logarithmic derivative of the current and noise on $V$ for various $u$. (b) The exponents at large bias voltages for various $u$. The parameters are $\epsilon /T_{\mathrm{K}}=0$, $t/\mathrm{\Delta }=0.001$, and $T/T_{\mathrm{K}}=0$.

Figure 7

The dependence of the noise on $V$ for (a) $T<T_{\mathrm{K}}$ and (b) $T>T_{\mathrm{K}}$ for various $u$ and $T$. The parameters are $\epsilon /T_{\mathrm{K}}=0$ and $t/\mathrm{\Delta }=0.001$.

Figure 8

The dependence of the equilibrium noise $S$ on $T$ for various $u$. The parameters are $\epsilon /T_{\mathrm{K}}=0$, $V/T_{\mathrm{K}}=0$, and $t/\mathrm{\Delta }=0.001$.

Figure 9

The dependence of the ratio between the noise and the backscattering current on $V$ for various $u$. The parameters are $\epsilon /T_{\mathrm{K}}=0$, $t/\mathrm{\Delta }=0.001$, and $T/T_{\mathrm{K}}=0$.

Figure 10

(a) The dependence of the third-order coefficient of the (a) current $G_3$ and (b) noise $S_3$ on $V$ for various $u$. The parameters are $\epsilon /T_{\mathrm{K}}=0$, $t/\mathrm{\Delta }=0.001$, and $T/T_{\mathrm{K}}=0$.

Figure 11

The vertex correction calculated using a FRG scheme and a plain perturbation theory for various $t$ as a function of $V$. The parameters are $u=0.1$, $\epsilon /T_{\mathrm{K}}=0$, and $T/T_{\mathrm{K}}=0$.

Figure 12

The vertex correction divided by $u^2$ calculated using our FRG scheme for various $u$ as a function of $V$. The parameters are $\epsilon /T_{\mathrm{K}}=0$, $t/\mathrm{\Delta }=0.001$, and $T/T_{\mathrm{K}}=0$.

Figure 13

(a) The current noise and (b) the vertex correction calculated from a plain perturbation theory for various $u$ and $t$ away from the particle-hole symmetric point as a function of $V$. The parameters are $\epsilon /T_{\mathrm{K}}=10$ and $T/T_{\mathrm{K}}=0$.

Figure 14

(a) The current noise and (b) the vertex correction divided by $u^2$ calculated using our FRG scheme for various $u$ and $t$ away from the particle-hole symmetric point as a function of $V$. The parameters are $\epsilon /T_{\mathrm{K}}=10$ and $T/T_{\mathrm{K}}=0$.

Figure 15

The dependence of the current noise on $V$ for various $u$ away from the particle-hole symmetric point. The parameters are $\epsilon /T_{\mathrm{K}}=10$, $t/\mathrm{\Delta }=0.01$, and $T/T_{\mathrm{K}}=0$.

Figure 16

The dependence of the current noise and its logarithmic derivative on $\epsilon$ for various $u$. The parameters are $V/T_{\mathrm{K}}=5$, $t/\mathrm{\Delta }=0.01$, and $T/T_{\mathrm{K}}=0$.