Universal low-temperature crossover in two-channel Kondo models

An exact expression is derived for the electron Green function in two-channel Kondo models with one and two impurities, describing the crossover from non-Fermi liquid (NFL) behavior at intermediate temperatures to standard Fermi liquid (FL) physics at low temperatures. Symmetry-breaking perturbations generically present in experiment ensure the standard low-energy FL description, but the full crossover is wholly characteristic of the unstable NFL state. Distinctive conductance lineshapes in quantum dot devices should result. We exploit a connection between this crossover and one occurring in a classical boundary Ising model to calculate real-space electron densities at finite temperature. The single universal finite-temperature Green function is then extracted by inverting the integral transformation relating these Friedel oscillations to the $t$ matrix. Excellent agreement is demonstrated between exact results and full numerical renormalization group calculations.

Figure 1

Schematic phase diagram for the 2CK and 2IK models as a function of temperature $T$, external energy scale $\omega$, and symmetry-breaking perturbation strength $\delta$. The three FPs of each model give rise to three distinct regimes: free fermion, NFL, and FL. We considered the NFL to FL crossover at $T=0$ in Ref. 26, indicated by arrow (a). Here, we generalize the results to finite temperature, arrow (b).

Figure 2

Schematic illustration of possible 2CK (a) and 2IK (b) setups to measure conductance. The left lead is “split” into source and drain, allowing the resulting conductance through the attached impurity to be measured.

Figure 3

Spectrum $t_{\sigma L}(\omega ,T)$ vs $\omega /T^{*}$ for $T/T^{*}={10}^{-1},1,10,{10}^2$, approaching $t_{\sigma L}=1/2$ from above (${\lambda }_1>0$) or below (${\lambda }_1<0$). Circles show $T=0$ result of Eq. (13).

Figure 4

$t_{\sigma L}(\omega =0,T=y)$ and $t_{\sigma L}(\omega =y\sqrt{b/a},T=0)$ vs $y/T^{*}$ for ${\lambda }_1>0$. Common FL asymptote Eq. (16) shown as dot-dashed line; NFL asymptotes Eq. (17) shown as dashed and dotted lines.

Figure 5

$t_{\sigma L}(\omega ,T)$ vs $\omega /T^{*}$ for $T/T^{*}={10}^{-1},1,10,{10}^2$, in the presence of finite uniform (staggered) magnetic field in the 2CK (2IK) model, ${\lambda }_B^z>0$. Spectra approach $t_{\sigma L}=1/2$ from above ($\sigma =\uparrow$) or below ($\sigma =\downarrow$). Circles show $T=0$ result.

Figure 6

Magnetization $[M\left(T\right)-M\left(0\right)]/\sqrt{T_K{T}^{*}}$ vs $T/T^{*}$ for ${\lambda }_B^z>0$ and $T_K/T^{*}={10}^6,{10}^7,{10}^8,{10}^9$. NFL asymptote Eq. (20b) shown as dashed line and FL asymptote Eq. (20a) shown as dotted line in the inset.

Figure 7

Conductance ${\tilde{G}}_c^L(V,T)\equiv G_c^L(V,T)/\left(2e^2{h}^{-1}{G}_0^L\right)$ vs bias $V$ and temperature $T$ for ${\lambda }_1>0$. Black lines connect regions of constant conductance. Light colors correspond to high conductance near the FL FP, dark colors correspond to lower conductance near the NFL FP.

Figure 8

Conductance ${\tilde{G}}_c^L(V,T)\equiv G_c^L(V,T)/\left(2e^2{h}^{-1}{G}_0^L\right)$ arising for finite ${\lambda }_1$. (Top) Conductance vs temperature $T/T^{*}$ for $V/T^{*}={10}^{-1},1,10,{10}^2$, approaching ${\tilde{G}}_c^L=1/2$ from above (${\lambda }_1>0$) or below (${\lambda }_1<0$). Circles show the exact zero-bias result. (Bottom) Conductance vs bias $V/T^{*}$ for $T/T^{*}={10}^{-1},1,10,{10}^2$. Diamonds show zero-temperature result.

Figure 9

A one-point function of a bulk operator evaluated at distance $x$ from the boundary in (a) is mapped to the two point function of the associated chiral fields in the absence of a boundary shown in (b) and evaluated at image positions with respect to the line $x=0$. (c) Generalization of the two point function away from image points.

Figure 10

Spectrum $t_{\sigma L}(\omega ,T)$ vs $\omega /D$ for the 2CK model with $\nu J=0.15$ and small channel anisotropy $2\nu {\Delta }_z={10}^{-6}$ at various temperatures $T/T^{*}={10}^{-1},1,10,{10}^2$, approaching $t_{\sigma L}=1/2$ from above. Dotted lines are results from full NRG calculations; solid lines are exact results from Eq. (9) for the NFL to FL crossover.

Figure 11

Impurity contribution to entropy $S_{\text{imp}}\left(T\right)$ vs $T/{\tilde{T}}^{*}$ for the 2CK model (upper panels) and the 2IK model (lower panels) in the presence of various perturbations. Entire temperature-dependence calculated by NRG for the full models (black lines); low-temperature $T\ll T_K$ behavior in each case compared with the single exact NFL–FL crossover function Eq. (101) (red circles). All results presented for $\nu J=0.25$. Left: effect of channel asymmetry $4\nu {\Delta }_z=\pm {10}^{-5}$ (2CK) or deviation from critical coupling $(K_c-K)/D=\pm {10}^{-5}$ (2IK). Center: effect of including also finite left/right tunneling, $2\nu {\Delta }_x=\pm {10}^{-5}$ (2CK) or $2\nu V_{LR}=\pm {10}^{-4}$ (2IK), with $4\nu {\Delta }_z=(K_c-K)/D=\pm {10}^{-5}$ as before. Right: effect of including finite magnetic field, $B^z/D=\pm {10}^{-9/2}$ (2CK) or $B_s^z/D=\pm {10}^{-9/2}$ (2IK), again with $4\nu {\Delta }_z=(K_c-K)/D=\pm {10}^{-5}$. Parameters chosen to allow direct comparison to Fig. 1 of Ref. 26.

Figure 12

Schematic illustration of a nonequilibrium 2CK setup. We consider the case of ${\Delta }_x\ll J$.

Figure 13

Scaling function for the nonlinear conductance of a 2CK device with small ${\Delta }_x$. From top to bottom at the peak: $T/T^{*}=0,0.25,0.5,1$.

Figure 14

Diagram for the inelastic contribution to the $t$ matrix. It describes interaction between two fermionic species $A\ne B$. In the 1CK model, $A,B\in \{\uparrow ,\downarrow \}$, in two-channel Kondo models $A,B\in \{\uparrow L,\downarrow L,\uparrow R,\downarrow R\}$.