1

URu2Si2 2

(n,l,m) + σ l: 4f (n=4,l=3) 5f (n=5,l=3) d 5 1. S 2. 1. L L=0(S), 1(P), 2(D), 3(F), 4(G), 5(H),... (2S+1) LJ 3

() d ~ > f >> > 1~10 ev 0.1~0.3 ev 1~100 K LS (Russell-Saunders) f 2 less than half-filled f 12 more than half-filled j-j << 4

f 1 (J=5/2) ρe(r) ρm(r) N S Jz 5

J=5/2 = + + + + + ( ) J=5/2 6

(-1) l cf. E-H (-1) l+1 = ( :N, :S) 7

(m>0) l 0 (monopole) 1 (dipole) 2 (quadrupole) 3 (octupole) 4 (hexadecapole) 5 (dotriacontapole) 6 (tetrahexacontapole) 8

f n (Wigner-Eckart) M, M! (x,y,z ) (Jx,Jy,Jz )! 9

Jx, Jy, Jz Jx, Jy, Jz Mathematica Stevens 10

(q~0) (C ) : () ) PrInAg2 : (Γ3) 10 K x 2 -y 2 O. Suzuki et al.: JPSJ 75 013704 '06 11

X X ( 1998 ) X ( ) 4f ψ 2p X cf. X 4f E1 E2 E1 ev 12

(X ) Ce 0.7 La 0.3 B 6 $(IV ) Mannix$et$al.$PRL$95$117206$ 05 L 2 $ $:$2p"5d$(E1),$2p"4f$(E2) T$=$1$K,$Q=(3/2,3/2,3/2) $(Γ 5u ) ψ σ"σ $$:$6 [111] HK$and$Y.$Kuramoto$JPSJ$74$3139$ 05 E2 13

( :) K.$Kuwahara$et$al.$JPSJ$76$(2007)$093702 R.$Shiina$et$al.$JPSJ$76$(2007)$094702 cαm(κ) F(κ) κ Q ( ) κ 14

cf. NMR/NQR xyz z xy y x 15

Hc > Hλ > HCEF J f r (Γ1g) = 16

(Oh) Oh J=5/2 4 O44 () 17

(Oh) () Γk ( ) = (2J+1)^2 ^2 18

Oh Γ3 J=4 19

Oh Γ3 4σz (Γ3 ) 4σx 20

Oh Γ3-18 sqrt(5) σy (Γ3 ) 40sqrt(15) σz 2 Γ3g () Γ3 Γ2u ( ) Γ3g ( ) 2 x 2 1 + 1 + 2 21

Oh Γ3 T. Onimaru et al.: PRL 106 177001 '11 Γ3 ()? 22

D4hx3 U 4+ (J=4) : UX2 (X=P, As, Sb, Bi), PrRu2Si2, UPt2Si2, URu2Si2 (?) J=4 x 3 URu2Si2 23

Oh Γ8 Ce 3+ (J=5/2) : Ce1-xLaxB6 ~ 540 K I. para II. AFQ (Γ5g) TRS III. AFQ (Γ5g) + AFM (Γ4u) IV. AFO (Γ5u) T. Tayama et al.: JPSJ 66 2268 '97 24

Th PrOs4Sb12 Th 4 Oh Γ4 Γ5 Γ4 (t) ~ 10 K Y. Aoki et al.: JPSJ 76 051006 '07 k=5 (θ~0) PrRu4P12 K. Iwasa et al.: PRB 72 024414 '05 AFQ: Γ5g SC AFH: Γ1g (scalar, hexadeca) 25

( ) 2 i j i j J ~ t 2 / U RKKY () i j m1 m2 m1 m2 26

2 (J<0) D(k) k (=Q) -4J D(k) χ(q) (0,0) (π,0) (π,π) (0,0) +4J 27

() (i, x) z : (m) 28

() (Q) T < TN T < TN (0) [ ](0) [] (Q) [] 29

CeB6 R. Shiina et al.: JPSJ 66 1741 '97 Γ3g Γ5g Γ4u Γ4u Γ5u Γ2u NMR M. Takigawa et al.: JPSJ 52 728 83 O. Sakai et al.: JPSJ 66 3005 97 X T. Matsumura et al.: PRL 103 017203 09 ξ(q) (001) (111) (110) σ (Q), η(q) 30

T > Tc T < Tc Γ1g 31

Oh3 Γ1g α = dipole (z), k=0 () O20(Q) Jz(Q) Oxy(Q) Txyz(Q) (C4v Γ4 ) 32

Ce1-xLaxB6 IV HK & Y. Kuramoto: JPSJ 70 1751 '01 K. Kubo & Y. Kuramoto: JPSJ 73 216 04 C44 ( ) 33

(2 ) k 0 () (Holstein-Primakoff ) HK and Y. Kuramoto JPSJ 70 3076 01 R. Shiina et al. JPSJ 72 1216 03 HK et al. JPSJ 78 094713 09 ( ) bi : HP b2 b1 34

T=0 α = 1 35

PrOs4Sb12 Y. Aoki et al.: JPSJ 76 051006 07 R. Shiina et al.: JPSJ 73 3453 04 Q 36

URu2Si2 R. Okazaki et al. Science 331 439 11 & T* M.B. Maple et al. PRL 56 (1986) 185 E. Hassinger, et al. PRB 77 (2008) 115117 37

URu2Si21. E. Hassinger, et al. PRB 77 (2008) 115117 D. Aoki et al. JPSJ 78 (2009) 053701 1st order (Ising ) T0, TN, Tx (HM, HAF) 38

URu2Si22. H. Amitsuka (Private commun.) G.J. Nieuwenhuys PRB 35 (1987) 5260 H c H a T > T* (T* ~ 50 K) H//c : CW H//a : VV Hc2 H. Ohkuni et al Phil. Mag. B 79 (1999) 1045 3D FS 39

URu2Si23. C. Broholm et al. PRB 43 (1991) 12809 T = 1 K Q=(1,0,0) Q=(1,0,0) ( Q=(100)) 40

URu2Si23. AFM A. Villaume et al. PRB 78 (2008) 012504 D. Aoki et al. JPSJ 78 (2009) 053701 0 0.5 1 GPa 41

URu2Si2 cf. Doniach T URu2Si2 Tcoh T* Torder T0 T* T0! QCP 0, AFM () 42

URu2Si2 Y. Kuramoto, 46 (1991) 98 Y. Kuramoto, Physica B 156&157 (1989) 789 NCA cf. DMFT N. Sato et al., JPSJ 54 (1985) 1923 T. Tayama et al., JPSJ 66 (1997) 2268 43

URu2Si2 (NMR) 29 Si NMR T. Kohara et al. Solid State Commun. 59 (1986) 603 5f (HF) 5f 44

URu2Si2 ( 2) G.J. Nieuwenhuys PRB 35 (1987) 5260 U 4+ (5f 2 ) J=4 3 H4 D4h ( ) 500 K 170 K PrRu2Si2 : R. Michalski et al. J. Phys. Condens. Matter 12 (2000) 7609 PrRu2Si2 : A. Mulders et al. PRB 56 (1997) 8752 UPt2Si2 : G.J. Nieuwenhuys PRB 35 (1987) 5260 RRu2Si2 (R=Th,Y,La) + Pr, U : A. Morishita et al. JMMM 310 (2007) 283 50 K cf. Haule-Kotliar LDA + DMFT [ Γ2 - Γ1 (1) ] Nature Phys. 5 (2009) 796; EPL 89 (2010) 57006 45

URu2Si2 170 K Jz> 0, 4 50 K z xy(x 2 -y 2 ) xyz(x 2 -y 2 )! 46

URu2Si2 z () xy(x 2 -y 2 ) () xyz(x 2 -y 2 ) z=0 z=c/2 47

URu2Si2 ( 1) (a) 1.5 (b) 1.5 T N (p) 1 T 0 (p) 1 H 0 (p) T / T 0 H / H 0 AFH 0.5 AFH AFM 0.5 AFM 0 0 0.5 1 1.5 p [GPa] 2 2.5 0 0 0.5 1 1.5 p [GPa] 2 2.5 48

URu2Si2 ( 3) (a) (b) ω ω ζ ζ ζ ζ ζ ζ ζ ζ (101) (100) (100) (110) (101) (100) (100) (110) Jz INS ξ : ξ INS Jz : Jx, Jy () 49

URu2Si2 ( ) Q=(1,0,0) :, H0 Q*=(1,0.4,0) :, cf. F. Bourdarot et al. PRL 90 (2003) 067203 P. Santini et al. PRL 85 (2000) 654 HK, arxiv:1108.4638 50

URu2Si2 X E2 H. Amitsuka, et al. J. Phys. Conf. Series 200 (2010) 012007 51

URu2Si2 AF x 2 -y 2 F H c [110] Oxy [110] O22 [100] Oxy RXS [110] 52