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 Γ₁, which models a Pr³⁺ ion with a non-Kramers double ground state Γ₃. We have used the numerical renormalization group approach and determined the phase diagram with varying quadrupole Kondo coupling J and Γ₁ excitation energy Δ. 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(Δ) and found a nice scaling behavior with an stretched exponential form T_K ∼ δJ^α exp(−const./\(\sqrt{\delta J} \)) where δJ ≡ J − J_c(Δ). This differs from the standard scaling form and indicates that one needs to consider renormalization of multiple coupling constants.