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

Abstract

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 $T_K\sim \delta J^{\alpha }\exp (-const./\sqrt{\delta J})$ 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.

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