Strong versus weak coupling duality and coupling dependence of the Kondo temperature in the two-channel Kondo model

We perform numerical renormalization group (NRG) as well as analytical calculations for the two-channel Kondo (2CK) model to obtain the dependence of the Kondo temperature $T_K$ on the dimensionless (bare) spin exchange coupling $g$ over the complete parameter range from $g⪡1$ to $g⪢1$. We show that there exists a duality between the regimes of small and large coupling. It is unique for the two-channel model and enables a mapping between the strong and the weak coupling cases via the identification $g\leftrightarrow 3∕\left(2g\right)$, implying an exponential dependence of $T_K$ on $1∕g$ and $g$, respectively, in the two regimes. This agrees quantitatively with our NRG calculations where we extract $T_K\left(g\right)$ over the complete parameter range and obtain a nonmonotonic $T_K\left(g\right)$ dependence, strongly peaked at the 2CK fixed point coupling $g^{*}$. These results may be relevant for resolving the long-standing puzzle within the 2CK interpretation of certain random defect systems as to why no broad distribution of $T_K$ is observed in those systems.

Figure 1

(Color online) Lowest energy levels of the isotropic 2CK model as a function of the number of NRG iterations $N$ ($N$ even) for different initial couplings $J$ and $\Lambda =4$ with 900 states kept at each iteration. As a guide to the eye, the levels obtained in the different NRG iterations $N$ are connected by straight lines. Independently of initial weak coupling $(J=0.2)$, intermediate coupling $(J=0.7)$, or strong coupling $(J=20)$ strengths, the same fixed point spectrum is reached. For odd number of iterations $N$ a nonequidistant fixed point spectrum is obtained as well (not shown).

Figure 2

(Color online) Dependence of the Kondo temperature $T_K$ on the bare coupling strength $J$ (both in units of $D$), as determined by NRG for $\Lambda =3$ and 4. The inset shows $T_K$ on a logarithmic scale.

Figure 3

(Color online) The Kondo temperature $T_K$ is shown on a logarithmic scale versus the parameter $-(1∕{\rho }_{0}J+\gamma {\rho }_{0}J)$, $\gamma =30∕46\approx 2∕3$. The upper branch of the curves corresponds to the weak coupling, the lower branch to the strong coupling regime. The results for $T_K$ obtained from the “inflection point method” (see text) in the weak coupling regime $(J<J^{*})$ are shown for comparison and differ only by a constant prefactor.