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 Pr3+ 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 TK near the phase boundary Jc(Delta) and found a nice scaling behavior with an stretched exponential form T-K similar to delta J(alpha) exp(-const./root delta J) where delta J equivalent to J- Jc(Delta). This differs from the standard scaling form and indicates that one needs to consider renormalization of multiple coupling constants.