Effect of ${\mathrm{\Gamma }}_7$ and ${\mathrm{\Gamma }}_8$ Hybridizations on Three-Channel Kondo Phase Emerging from Ho Ions

By employing a numerical renormalization group method, we analyze a seven-orbital impurity Anderson model for Ho${}^{3+}$ ion with ten $4f$ electrons. This model includes both $V_7$ and $V_8$, which are hybridizations between localized $4f$- and conduction electrons in ${\mathrm{\Gamma }}_7$ and ${\mathrm{\Gamma }}_8$ orbitals, respectively. For the case of $V_7=V_8$ with the local ${\mathrm{\Gamma }}_5$ triplet ground state, we have reported the discovery of a three-channel Kondo (TCK) phase, characterized by a residual entropy of $\log \phi$ with the golden ratio $\phi =(1+\sqrt{5})\mathrm{/}2$. In this research, by depicting the ground-state phase diagram on the $(V_8,V_7)$ plane, we attempt to unveil the effect of $V_7$ and $V_8$ on the emergence of the TCK phase. After performing a lot of numerical calculations, we find that the TCK phase appears in a relatively wide region on the $(V_8,V_7)$ plane. The boundary curves surrounding the TCK phase are determined by the variation of the temperature dependence in entropy and the abrupt change in energy spectra. We consider that most of the phases surrounding the TCK phase are Fermi liquids, but the non-Fermi liquid two-channel Kondo phase is unexpectedly found to exist next to the TCK phase. Finally, we briefly comment on the actual material concerning the detection of the TCK phase.