Stable Two-Channel Kondo Fixed Point of an SU(3) Quantum Defect in a Metal: Renormalization-Group Analysis and Conductance Spikes

We propose a physical realization of the two-channel Kondo (2CK) effect, where a dynamical defect in a metal has a unique ground state and twofold degenerate excited states. In a wide range of parameters the interactions with the electrons renormalize the excited doublet downward below the bare defect ground state, thus stabilizing the 2CK fixed point. In addition to the Kondo temperature $T_K$ the three-state defect exhibits another low-energy scale, associated with ground-to-excited-state transitions, which can be exponentially smaller than $T_K$. Using the perturbative nonequilibrium renormalization group we demonstrate that this can provide the long-sought explanation of the sharp conductance spikes observed by Ralph and Buhrman in ultrasmall metallic point contacts.

Figure 1

(a) Level scheme of the three-state defect, defining the coupling constants in Eq. (1); (b) 2nd and (c) 3rd order contributions to the current through a point contact with Kondo impurities. Solid line: conduction electron; dashed line: defect state propagators. A ● indicates a sum over all 4-point vertices of Eq. (1).

Figure 2

Coupling function $g_{01}^{(0)}(\omega )$ for different bias voltages $V$ (parameter values as in Fig. 3). The peak structure of $g_{01}^{(0)}(\omega )$ is centered around the renormalized level spacing $\Delta (D=0)$ and is split by the respective bias $V$. The inset shows the defect self-energy ${\Sigma }_m(\omega )$, $m=0$, 1, calculated in bare 2nd order PT. The level-spacing renormalization is the difference between the two solid dots, $\delta \Delta =\mathrm{Re}[{\Sigma }_1(\Delta )-{\Sigma }_0(\Delta )]$.

Figure 3

Phase diagram in the $J_{\perp }^2-{\Delta }_0$ plane for $J_z/2=J_{\perp }=0.018$, bias $V=0$, and various values of $g_{0m}^{(n)}=g_{m0}^{(n)}$, $n$, $m=0$, $\pm 1$. The symbols mark the line of instable fixed points separating the 2CK (left of the line) from the potential scattering (right of the line) low-temperature phase. The solid lines are straight line fits to the $J_{\perp }^2$ vs ${\Delta }_0$ plots and show that the level crossing is essentially an $O(J_{\perp }^2)$ PT effect. The inset shows typical RG flow diagrams of the level spacing $\Delta (D)$ in (a) the 2CK phase (level crossing) and (b) the potential scattering phase (no level crossing).

Figure 4

Differential conductance correction $dI/dV$ per defect for bias $V$ near the renormalized level spacing $\Delta (D=0)$, $V>T_K\approx {10}^{-4}{D}_0$. $J_z/2=J_{\perp }=0.016$, $g_{01}^{(j)}=g_{10}^{(j)}=0.6J_{\perp }$, ${\Delta }_0=3T_K$, $\Delta (D=0)=-1.82T_K$. Right inset: $dI/dV$ for a larger bias range. The dashed line is a sketch of the 2CK-induced $\sqrt{|V|}$-ZBA of width $T_K$, with the numerically determined $T_K$, as indicated. Left inset: experimental $dI/dV$ curve taken from Ref. 11 for comparison.