Numerical Renormalization Group Study of Quadrupole Kondo Effect with the Crystal-field Excited State

We have studied the quadrupolar Kondo effect for an impurity in cubic environment with taking account of a singlet excited state $\Gamma_1$, which models a $\mathrm{Pr}^{3+}$ ion with a non-Kramers double ground state $\Gamma_3$. We have used the numerical renormalization group approach and determined the phase diagram with varying quadrupole Kondo coupling $J$ and $\Gamma_1$ excitation energy $\Delta$. Two phases are found and identified as local Fermi liquid and non-Fermi liquid. This non-Fermi liquid phase is characteristic to the two-channel Kondo impurity, and a similar phase diagram has been also found in other extended quadrupole models. We have analyzed in detail the $J$-dependence of the Kondo temperature $T_K$ near the phase boundary $J_c (\Delta)$ and found a nice scaling behavior with an stretched exponential form $T_K \sim \delta J^{\alpha} \exp (-\mathrm{const.}/\sqrt{\delta J})$ where $\delta J \equiv J - J_c (\Delta)$. This differs from the standard scaling form and indicates that one needs to consider renormalization of multiple coupling constants.