Dynamical conductance in the two-channel Kondo regime of a double dot system

We study finite-frequency transport properties of the double dot system recently constructed to observe the two-channel Kondo effect [R. M. Potok et al., Nature 446, 167 (2007)]. We derive an analytical expression for the frequency-dependent linear conductance of this device in the Kondo regime. We show how the features characteristic of the two-channel Kondo quantum critical point emerge in this quantity, which we compute using the results of conformal field theory as well as numerical renormalization group methods. We determine the universal crossover functions describing non-Fermi-liquid vs Fermi-liquid crossovers and also investigate the effects of a finite magnetic field.

Figure 1

(Color online) Two-dot device: the small dot in the center couples to a large dot (2) and to a left and a right lead ($1L$ and $1R$) via the hopping amplitudes $v_L$ and $v_R$. The small dot has a large level spacing, and the large dot is characterized by a vanishing level spacing, while both dots are in the Coulomb blockade regime. As a result, only spin exchange is possible between the dots.

Figure 2

(Color online) Top: Sketch of the conductance through the small dot divided by its maximum value, $G_0$, as a function of temperature. For $J_1=J_2$, $a\sim \sqrt{T}$ singularity emerges, while for $J_1\ne J_2$, a Fermi liquid is formed at a scale $T^{*}$, and the conductance crosses over to a very small or a large value, with a characteristic Fermi-liquid scaling, $\sim {(T∕T^{*})}^2$. Bottom: Sketch of the “phase diagram” of the two-channel Kondo model.

Figure 3

(Color online) Imaginary part of the eigenvalue of the $T$ matrix obtained by numerical integration of Eq. (22). The scale of the $\omega$ axis is set by the amplitude of the leading irrelevant operator, $\lambda$. The inset illustrates how the curves corresponding to different temperatures collapse into one universal curve.

Figure 4

(Color online) Real part of the eigenvalue $t\left(\omega \right)$ of the $T$ matrix predicted by conformal field theory. The inset shows the collapse to a single scaling curve (obvious from the integral definition).

Figure 5

(Color online) Real part of the conductance computed from Eqs. (22, 16, 17). The inset shows the universal collapse.

Figure 6

(Color online) Imaginary part of the conductance from Eqs. (22, 16, 17). The inset shows the universal scaling curve.

Figure 7

(Color online) (a) Imaginary part of the eigenvalue of the on-shell $T$ matrix, as function of $\omega ∕T_K$, for several different values of the anisotropy parameter, $K_R=4(J_1-J_2)∕{(J_1+J_2)}^2$. In all cases, $J_1+J_2=0.2$. Curves with $J_1>J_2$ or $J_1<J_2$ scale to $\mathrm{Im}t\left(0\right)=2$ or $\mathrm{Im}t\left(0\right)=0$, respectively. The critical curve corresponding to $J_1=J_2$ separates these two sets of curves. (b) $\mathrm{Im}t\left(\omega \right)$ for $J_1=J_2$ as a function of $\sqrt{\omega ∕T_K}$. The dashed line is a guide for the eye. (c) $T^{*}$ as a function of $K_R^2$.

Figure 8

(Color online) Imaginary part of the on-shell $T$ matrix in the presence of channel anisotropy as a function of $\omega ∕T^{*}$. The upper part corresponds to $J_1>J_2$, while the lower part to $J_1<J_2$. In both cases, for $T^{*},\omega ⪡T_K$, the curves follow the universal crossover function, corresponding to a ${(\omega ∕T^{*})}^2$-like scaling at low frequencies and a $1\pm c{(T^{*}∕\omega )}^{1∕2}$ behavior at large frequencies.

Figure 9

(Color online) (a) ac conductance as a function of $\omega ∕T_K$. For $J_1>J_2$ and $J_1<J_2$, the curves scale as $\mathrm{Re}G\to G_0$ and $\mathrm{Re}G\to 0$, respectively. Inset: ac conductance for $J_1=J_2$ as a function of $\sqrt{\omega ∕T_K}$. (b) ac conductance for positive (upper part) and negative (lower part) channel anisotropy parameters as a function of $\omega ∕T^{*}$. For $\omega ,T^{*}⪡T_K$, the curves follow the universal crossover curves.

Figure 10

(Color online) Imaginary part of the ac conductance as a function of $\omega ∕T_K$. Lower panel: Same as that of the upper panel but as a function of $\omega ∕T^{*}$.

Figure 11

(Color online) Top: Imaginary part of the on-shell $T$ matrix in the presence of a magnetic field and no channel asymmetry, as a function of $\omega ∕T_K$. Lower curves correspond to higher magnetic fields. Bottom: ac conductance in the presence of a magnetic field and no channel asymmetry, as a function of $\omega ∕T_K$. Lower curves correspond to higher magnetic field values.

Figure 12

(Color online) Sketch of the real and imaginary parts of the ac conductance for $J_1=J_2$ and $\omega ,T⪡T_K$.

Figure 13

(Color online) Sketch of the real and imaginary parts of the $T=0$ ac conductance for $J_1>J_2$. The various powers shown in the picture follow from conformal field theory. The high frequency behavior is a result of perturbation theory. We assumed electron-hole symmetry.