ˆ Q$_{xz}$ = c$_{Q}$ $^{[(}$| $^{Γ$_{1}$}$〉〈Γ + 5 | + | Γ + 5 〉〈 $^{Γ$_{1}$}$| ) − $^{(}$| $^{Γ$_{1}$}$〉〈Γ − 5 | + | Γ − 5 〉〈 $^{Γ$_{1}$}$| )] , (2) ˆ Q$_{yz}$ = − i c$_{Q}$ $^{[(}$| $^{Γ$_{1}$}$〉〈Γ + 5 | − | Γ + 5 〉〈 $^{Γ$_{1}$}$| ) − $^{(}$| $^{Γ$_{1}$}$〉〈Γ − 5 | − | Γ − 5 〉〈 $^{Γ$_{1}$}$| )] . (3) H$_{K}$ = J$_{Q}$ ( ˆ Q$_{xz}$ ˆ q$_{xz}$ + ˆ Q$_{yz}$ ˆ q$_{yz}$ ) . (4) The key point remains to observe the symmetry lowered in HO. As many unsuccessful attempts have been tried; obvi- ously improvements in the method’s sensitivity are necessary. For example careful check of the anisotropy of the magneti- zation in the basal plane above T$_{O}$ will be worthwhile as short correlations of the quadrupole pair ( O$_{xy}$ - $^{O}$− $_{xy}$) must enhance a 4 fold component. Refinement in resonant X ray measure- ments (well-known tool in the proof of quadrupole ordering) is required as the last result fails to detect any quadrupole or- In higher pressure phase, large moment antiferromagnetism (LMAF) of Ising type (with c -axis as an easy axis) is realized. In order to treat the LMAF and HO order on an equal footing, we have to extend the Hilbert space, spanned by | $^{Γ$_{1}$}$〉 and | Γ ± 5 〉 , by adding the CEF state Γ$_{2}$ (A$_{2g}$; xy ( x 2 − y $^{2}$)) whose eigen function is | $^{Γ$_{2}$}$〉 = $^{(}$|$^{4}$〉 −  | − $^{4}$〉) / √$_{2. The detailed discussions}$ are discussed in a separate paper. A Nambu-Goldstone (NG) mode around this HO state is expected to consist of doubly degenerate excited states Γ ± 5 and the singlet ground state Γ$_{1}$ , as in the case of “gap- less spin waves” in CEF singlet systems such as Pr-based compounds. 31) Quite recently, two characteristic phenomena which seem to have a potentiality to unveil the form of HO were reported. One is resistivity measurements at low temperature region: the resistivity along the a -axis, ρ$_{a}$ , ex- hibits a non-Fermi liquid behavior, ρ$_{a}$ = ρ$_{0}$$_{a}$ + A$_{a}$T , and that along the c -axis shows the usual Fermi liquid behavior, ρ$_{c}$ = ρ$_{0}$$_{c}$ + A$_{c}$T $^{2}$.32) Another one is neutron scattering measurement: it was reported on neutron scattering experiment at P = 0 . 67 GPa that an inelastic mode of magnetic fluctuations m$_{z}$ at the wave vector at Q = (1 , 0 , 0) exists only in the HO phase, not in the normal nor the large moment antiferromagnetic phases. 19) These two unusual phenomena can be explained if the NG mode in HO state is a linear combination of quadrupolar fluc- tuation of Q$_{xz}$ and Q$_{yz}$ representing Γ ± 5 state in tetragonal sym- metry. 33) The detailed analysis is now in progress. where ˆ Q$_{xz}$ (ˆ q$_{xz}$ ) and ˆ Q$_{yz}$ (ˆ q$_{yz}$ ) are quadrupole coordinates of U- ion with f $^{2}$-configuration (of conduction electrons described by the symmetry of f $^{1}$-configurations). This interaction can give rise to an “induced-moment Kondo e ff ect” as in the case of Pr-based skutterudite compounds. 30) Detailed discussions will be given in a separate paper. These two operators form the SU(2) Lie algebra together with their comutator. Corresponding to the quadrupole oper- ator ˆ Q$_{xz}$ and ˆ Q$_{yz}$ in the f $^{2}$-configuration, we can define the quadrupole operators ˆ q$_{xz}$ and ˆ q$_{yz}$ in the f $^{1}$-configuration. Then, from a symmetry argument, there exists a Kondo type inter- action We thank H. Amitsuka, K. Behnia, Y. Matsuda, T. Sakak- ibara, J.A. Mydosh, A, Kiss, C. Pepin, T. Mito and Z. Hiroi for useful and fruitful comments. We are pleased to dedicate this paper to Patrik Fazekas, for his great service to this fie ld. HH also thanks Aspen Center for Physics for his two week stay on the related subject. This work was partly supported by a Grant-in-Aid for Scientific Research on Innovative Areas ”Heavy Electrons” (No. 20102002) of The Ministry of Edu- cation, Culture, Sports, Science, and Technology, Japan. Q$_{yz}$ ≡ { J$_{y}$ , $^{J$_{z}$}$} / 2 between Γ$_{1}$ and Γ ± $_{5}$: i.e., 〈 $^{Γ$_{1}$}$| $^{Q$_{xz}$}$| Γ ± 5 〉 = ± c$_{Q}$ , and 〈 $^{Γ$_{1}$}$| $^{Q$_{yz}$}$| Γ ± 5 〉 = ∓ i c$_{Q}$ , with c$_{Q}$ ≡ $_{(3}$√$_{2}$$_{αε}$ − √$_{5}$$_{βγ}$$_{). The HO}$ of No.136 can be identified as an induced-moment AF order of Q$_{xz}$ + Q$_{yz}$ at A sublattice points and $^{Q$_{xz}$}$− Q$_{yz}$ at B sublattice points, respectively. Note that the symmetry of charge distri- bution is O$_{xy}$ : [ z ( x ± y )] 2 = z $^{2}$( x 2 + y 2 ± 2 xy ). In the Hilbert space spanned by Γ$_{1}$ and Γ ± $_{5}$, the quadrupole operator ˆ Q$_{xz}$ and ˆ Q$_{yz}$ are represented as der. 34) Further experimental e ff orts are greatly encouraged. 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