Two-Channel Kondo Physics in Two-Impurity Kondo Models

We consider the non-Fermi-liquid quantum critical state of the spin-$S$ two-impurity Kondo model and its potential realization in a quantum dot device. Using conformal field theory and the numerical renormalization group, we show the critical point to be identical to that of the two-channel Kondo model with additional potential scattering, for any spin $S$. Distinct conductance signatures are shown to arise as a function of device asymmetry, with the square-root behavior commonly believed to arise at low-energies dominant only in certain regimes.

Figure 1

Spectra $D{\rho }_{\alpha }(\omega )$ versus frequency $|\omega |/D$ for channels $\alpha =L$ and $R$ (upper and lower panels, respectively) for the asymmetric 2IKM with fixed Kondo exchanges $\rho J_L=0.1$, $\rho J_R=0.05$, varying interimpurity exchange $K$ (and $H_{\mathrm{ps}}=0$). Plotted using $K=K_c(1\pm {10}^{-n})$ for dashed lines and solid lines, with integer $n=1\to 6$ approaching progressively the critical point $K_c\sim T_K^L\approx {10}^{-6}D$ (dotted line). Circles: A 1CK model with $\rho J=\rho J_L$. Diamonds: $D{\rho }_{2\mathrm{CK}}(\omega )$ for a pure 2CK model with $T_K^{2\mathrm{CK}}=T_c$. Squares: $D{\tilde{\rho }}_{2\mathrm{CK}}(\omega )=1-D{\rho }_{2\mathrm{CK}}(\omega )$.

Figure 2

$T=0$ conductance $G_{2\mathrm{IK}}^{\alpha }/(2e^2{h}^{-1}{G}_0^{\alpha })$ through channel $\alpha =L$ and $R$ (solid and dashed lines) versus bias $V_{sd}/D$, at the critical point. Shown for $\rho J_L=0.075\ge \rho J_R$, varying $\rho J_R=0.075\to 0.05$ in steps of 0.0025, with $K=K_c\sim T_{KL}^{1\mathrm{CK}}$ retuned in each case (and $H_{\mathrm{ps}}=0$). Thick solid line is the symmetric case $J_L=J_R$; asymmetry $J_L/J_R\ge 1$ increases in direction of arrows. Circle and diamonds: Pure 1CK and 2CK scaling spectra. Inset: Zero-bias conductance versus $T/T_c$ for $J_L/J_R=1$ and 2 (and $\alpha =L,R$), exhibiting, respectively, leading linear and square-root behavior (dotted lines).

Figure 3

Impurity entropy $S_{\mathrm{imp}}(T)$ versus $T/D$ for the 2IKM with spin-1 impurities. Plotted for fixed $\rho J_L=0.15$, $\rho J_R=0.05$, varying $K/D=0,{10}^{-13},3\times {10}^{-7}$, $K_c\approx 6\times {10}^{-5}$, $2\times {10}^{-4}$, and ${10}^{-2}$ for lines (a)–(f). The corresponding physical processes are illustrated in panels (a)–(f): impurities denoted as circles and conduction band orbitals (in the Wilson chain representation [18]) as squares. For discussion, see text.