S=1 Bilinear-Biquadratic
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1 S=1 Bilinear-Biquadratic
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3 i 1983 Haldane S=1 1 Haldane Haldane Affleck-Kennedy-Lieb-Tasaki(AKLT) Valence Bond Solid(VBS) VBS (AKLT ) S=1 Bilinear-Biquadratic 1 Haldane Haldane 1 Haldane AKLT (AKLT ) ( ) S=1 Bilinear-Biquadratic ( ) VBS S=1 Bilinear-Biquadratic 1 Haldane Haldane Uimin-Lai-Sutherland Unknown I Berezinskii-Kosterlitz-Thouless(BKT) massless
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5 iii i S= 1 XXZ Haldane S=1 Bilinear-Biquadratic S=1 Bilinear-Biquadratic S=1 Bilinear-Biquadratic VBS S= S=1 Bilinear-Biquadratic S=1 Bilinear-Biquadratic S=1 Bilinear-Biquadratic S=1 Bilinear-Biquadratic Uimin-Lai-Sutherland Uimin-Lai-Sutherland Uimin-Lai-Sutherland SU(3)
6 iv 7.3 [ ] Unknown I A KOBPACK 83 A.1 KOBPACK A A.3 Householder Lanczos A.4 Lanczos A.5 KOBPACK B 93 B.1 S= B. S= C 97 C C C C C C C.7 A
7 v.1 Mermin-Wagner Berezinskii-Kosterlitz-Thouless Haldane S=1 Bilinear-Biquadratic [56] (6.16)(6.17) S=1 (6.16)(6.17) Unknown I A.1 KOBPACK A. Lanczos A.3 Lanczos C.1 (k) C. f A (k) = J sin(k) C.3 f B (k) = J J cos(k) C.4 f A (k) f B (k) C.5 J > 0 Ė(k) (k) C.6 J < 0 (k) (k)
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9 vii.1 Ψ (N) αβ S= VBS VBS (a) (b) S=1 BLBQ [43] N VBS p p up p down KOBPACK (a) ξ(α) (b) q IC (α) L α ST [8] ( 1 3 < α < 1) q IC(α) H p,β H p,β S=1 BLBQ P=0 ( 4.6 ) q IC (β) P=0.06 ( 4.6 ) q IC (β) P=0.15 ( 4.6 ) q IC (β) P=0.1 ( 4.6 ) q IC (β) P=0.33 ( 4.6 ) q IC (β) P=-0.06 ( 4.6 ) q IC (β) P=-0.15 ( 4.6 ) q IC (β) P=-0.1 ( 4.6 ) q IC (β) Unknwon L=1 F (0) L=1 F (1) F (0) F (1) L=1 F (10) L=1 F (11) F (10) F (11) L=9 triplet-quintet L=1 triplet-quintet
10 viii 6.3 L=15 triplet-quintet L=18 triplet-quintet p cross (β = 0.66) = a + b 1 L p cross (β = 0.68) = a + b 1 L L=9,1,15 fit β=-0.64 ( 7.1 ) c β= 3 ( 7.1 ) c β=-0.70 ( 7.1 ) c L=1 (1) L=1 () L=1 (3) L=1 (4) L=1 (5) L=1 (6) L=1 (7) L=1 (8) L=1 (9) L=1 (10) L=1 (11) L=15 (1) L=15 () L=15 (3) L=15 (4) L=15 (5) L=15 (6) L=15 (7) L=15 (8) L=15 (9) L=15 (10) L=15 (11) L=15 (1) L=15 (13) L=15 (14) Haldane A.1 KOBP ACK A. AV = V C.1 G = Sym[ m ν m,m C. ν i,j C.3 m ν m,m
11 ix C.4 α i,j = Sym[ ν i,j ν i 1,j+1 m i 1,j ν m,m+1 ] C.5 β i,j = Sym[ ν i 4,j+3 ν i 3,j+ ν j+1,i ν j,i 1 m i,i 1,j,j+1 ν m,m+1 ] C.6 i 1, M 1 ; i, M α C.7 i 1, M 1 ; i, M β C.8 J, J A,B,C,D,,F C.9 C.10 C.18 (J,J ) C.10 A J = 0, J = C.11 B J = 1, J = C.1 C J = 1, J = C.13 D J = 1, J = C.14 J = 0.1, J = C.15 F J = 0.1, J = C.16 G J = 1, J = C.17 H J = 1, J = C.18 I J = 1, J =
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13 Coulomb ( ) Hall ( ) [1] ( *1 ) *1
14 [1] 1930 Bethe Bethe S = 1 1 * Bethe( ) 1 S = 1 ( ) 1. ( ). ( ) 3. * 3 ( ) ( ) S = 1 1 S Haldane [] 1 S (S = ) 1 3 (S = 1 3 ) 1.. (Haldane gap) ( ) 3. * 4 S=1 Ni(C H 8 N ) NO(ClO 4 )(NNP)[3] [4] (CH 3 ) 4 NNiNO 3 (TMNIN) [5] Y MaNiO 5 [6] [7] 1 Affleck-Kennedy-Lieb-Tasaki[8] [9](AKLT) (valence bond solid,vbs) VBS (AKLT ) [10] AKLT 1 1 S=1 Bilinear-Biquadratic * Bethe *3 S i S j = i j 1 ( ) -1 *4 ξ( ) S i S j = exp( i j ) ξ
15 S=1 Bilinear-Biquadratic ( S=1 BLBQ ) AKLT (AKLT ) Haldane 1 Haldane AKLT (VBS ) S0S z n z = ( 1) n exp( n /ξ(α)) S0S z n z ( ) S=1 BLBQ ( ) VBS S=1 Bilinear-Biquadratic ( S=1 BLBQ ) S=1 BLBQ AKLT (VBS )1 VBS 1 ( S=1 BLBQ ) S=1 BLBQ ( S=1 BLBQ 1 H p,β ) Haldane Haldane Uimin-Lai-Sutherland Unknown I Berezinskii-Kosterlitz-Thouless(BKT) massless 1.3 R (eview) A (nalysis) F (ind) 1 1. R. R 1 a S= 1 S= 1 XXZ b S=1 Haldane c Haldane VBS VBS AKLT d AKLT 1 1 S=1 Bilinear-Biquadratic 3 S=1 Bilinear-Biquadratic R S=1 Bilinear-Biquadratic 4 1. R S=1 Bilinear-Biquadratic ( Haldane ). R S=1 Bilinear-Biquadratic ( )
16 A S=1 Bilinear-Biquadratic 1 H p,β 1 H p,β 4. F 1 H p,β Haldane Unknown I Unknown II 5 1. R. A 1 H p,β ( ) 3. F 1 H p,β 6 Uimin-Lai-Sutherland 1. R Unknown II S=1 Bilinear-Biquadratic Uimin-Lai-Sutherland. A Uimin-Lai-Sutherland 3. F 1 H p,β Uimin-Lai-Sutherland ( ) 4. R R. R q 3. A 6 4. F 6 BKT Unknown II Berezinskii-Kosterlitz-Thouless(BKT) massless 8 Uknown I A F Unknown I 9 1 H p,β
17 5 ( ) * ( ) 1 [11] d d S= 1 Bethe S= 1 XXZ S 1 Haldane ( ) (valence bond solid VBS) VBS AKLT AKLT 1 1 S=1 Bilinear-Biquadratic S= 1 H = L S i S i+1 = i=1 L i=1 1 ( S + i S i+1 + S i i+1) S+ + S z i Si+1 z *1.1.3
18 6 ( ) Neel Neel = i 1 i ( ( ) ) ( ) ( )Neel S i [S i, S j ] = iɛ ijk S k (i, j, k = 1,, 3) ɛ ijk : * S + i S i S= 1 1 S+ i S i+1 Neel S + i S i+1 Neel = S+ i S i+1... i 1 i i+1 i+... =... i 1 i i+1 i+... Neel ( )Neel ( )Neel Neel S= 1 [10].1 1 [1] ( 1 ).1.1 H = S i S j (.) ij ij i j SO(3) SU() XY 1 xy H XY = Si x Sj x + S y i Sy j (.3) ij * ɛ ijk 8 >< 1 ((i, j, k) = (1,, 3) (, 3, 1) (3, 1, )) ɛ ijk = 1 ((i, j, k) = (1, 3, ) (3,, 1) (, 1, 3)) >: 0 (.1)
19 .1 7 XY z SO() U(1) z S z H Ising = σ i σ j (.4) ij σ i ±1 σ i Z XY 3 3 XXZ H XXZ = Si x Si+1 x + S y i Sy i+1 + Sz i Si+1 z (.5) ij > 1 > 1 < 1 XY < 1 XY = 1 3 Z SO() SO(3)/SO() 1.1. SO(3) Z.1.3 Mermin-Wagner Berezinskii-Kosterlitz-Thouless SO(3) SO() 1 (T 0) Mermin-Wagner [13][14][15] 1 XY (T = 0)? 1 ( )
20 8 1 1 ( ) [16] Mermin-Wagner XY 0 Berezinskii-Kosterlitz-Thouless(BKT) SO()(U(1)) ( 1 U(1) ) [17][18].1 Mermin-Wagner Berezinskii-Kosterlitz-Thouless (T 0) (T = 0) SO(3) 1 A B SO() 1 A XY B Mermin-Wagner A B Berezinskii-Kosterlitz-Thouless SO()(U(1)) 1 U(1). 1 S= 1 XXZ 1 S= XXZ H XXZ = J N ( S x i Si+1 x + S y i Sy i+1 + Sz i Si+1) z i (.6) ( (S N+1 = S 1 ) N ) (.6) 1 S z π J J J>0 J<0 J>0
21 . 1 S= 1 XXZ 9 Jordan-Wigner S z = n i 1 S + i S i = c i Πi 1 j=1 (1 n j) = c i Π i 1 j=1 (1 n j) (.7) S= 1 c i c i i { } c i, c { } { } j = δ ij ci, c j = c i, c j = 0 (.8) n i c i c i (.7) (.7) (.6) H XXZ = J N i {( ) c i c i+1 + c i+1 c i + ( ( n i ) 1 ni+1 ) } 1 (.9) * S= 1 XY [ = 0] = 0 XXZ x y XY (.9) 1 Π i 1 j=1 (1 n j) n i = 1 n i n i *4 k m ( πm N ) ɛ km = J cos k m (.10) *5 ɛ km < 0 π < k < π 1 Si α (α = x y z) 0 = π < k m <π J cos k m = J N π π π cos kdk = NJ π i j 1 (.11) S x i S x j ( 1)i j i j 1 S z i S z j (1 ( 1)i j ) i j 1 (.1) (.13) (.14) *3 C C.1.1 *4 C C.1. *5 C C.1.3
22 10 [19][0] xy *6 S z 1 k < π 1 Sz 1 k > π 1 k > π k < π S Z k S Z J sin k < ω < J sin 1 (.15) 1 S= 1 XY 1 z (.9) 0 S= 1 XXZ 1 Bethe Bethe [1][][3].. [ > 1] > 1 S z M *7 s 1 1 π { M s ( 1) exp π ( 1) 0 } (.17)..3 [ = 1] Hulthen[1] Bethe 1 1 J(log 1 4 ) 0.443J S α i S α j ( 1)i j (log i j ) 1 i j (α = x, y, z) (.18) [4][5] *6 C C.1.3 *7 M s = 1 X e iq r i S i (.16) N i (staggered magnetization) r i i Q
23 .3 Haldane XY [ < 1] 1 XY η S x i S x j ( 1)i j i j η (.19) η = arcsin (.0) π = 1 = S z ( ) S z i S z j ( 1)i j i j 1 η (.1) η < 1 XY [6][7]..5 [ < 1] S z S z = ± S= 1 XXZ Berezinsukii-Kosterlitz-Thouless.1.3 XY BKT XY 1 S= 1 XXZ = 1 < 1 ( ) BKT *8 (massless) ( 1 < 1) c = 1 *9.3 Haldane.3.1 Haldane S = 1 S 1 S S = F.D.M.Haldane (S = 1, 3,... ) S = 1 *8 masslee *9 7
24 1 (S = 1,... ) [] Haldane S = 1 Haldane Haldane Haldane [1].3. AKLT Haldane Affleck-Kennedy-Lieb-Tasaki[8] [9](AKLT) (valence bond solid, VBS) AKLT VBS (AKLT ) AKLT S = 1 1 H AKLT = i P (S i + S i+1 ) (.) P (S i + S i+1 ) S i + S i+1 ( ) S = 1 1 S i + S i+1 = 1 S i S i+1 = 1 (.3) 0 P (S i + S i+1 ) = 1 (S i S i+1 ) (S i S i+1 ) (.4) * 10 AKLT (.) S = 1 S = 1 S = 1 Ψ i.αβ = 1 [ ψ (i,1) α ψ (i,) β + ψ (i,1) β ψ α (i,) ] (.5) i S = 1 ψ (i,j), ψ (i,j) i S = 1 j (j = 1,) Ψ i.αβ S = 1 S z j 1 i, 0 i, 1 i Ψ i. = 1 i, Ψ i. = Ψ i. = 0 i, Ψ i. = 1 i (.6) S = 1 S = 1 S = Ψ 1.α Ψ. β Ψ 1.α Ψ. β = Ψ 1.αγ ɛ γδ Ψ.δβ (.7) *10 C C.1.4
25 .3 Haldane 13 * 11 ɛ γδ (ɛ γδ = ɛ δγ, ɛ = 1) ( (.7)) α, β, S = α β, N Ψ (N) αβ = Ψ 1.αβ 1 ɛ β 1α Ψ.α β ɛ β α 3... ɛ β N 1α N Ψ N.αN β (.8) ( (.)) α β, Ψ (N) i.αβ Ω (N) = Ψ (N) αβ ɛβα (.9) ( ) 1 N-1 N... α β.1 Ψ (N) αβ.1 S = 1 S = 1 (valence bond) (valence bond solid, VBS) (S = 1 S = 1 Haldane ) VBS i [ ] [ ] [ ] Ψi. Ψ g i = i. 0 1 Ψi. Ψ = i. 1 0 Ψ i. Ψ i. Ψ i. Ψ i. (.30) N VBS Ψ (N) i.αβ g i N g i = i=1 [ ] [ 0 1 Ψ (N) 1 0 Ψ (N) * 1 VBS [ N ] Ω (N) = T r g i i=1 Ψ (N) Ψ (N) Ψ i.αβ (ket bra) Ψ i.αβ ] [ Ψ g i = i. Ψ i. Ψ i. Ψ i. ] (.31) (.3) (.33) *11 C C.1.5 *1 C C.1.6
26 14 Ω (N) Ω (N) = ( N ) g i i=1 N g j αα j=1 ββ = T r [ G N ] (.34) * 13 G 4 4 G αβ;γδ = g i,αγ g i,βδ (.35) (.35) G i Ω (N) Si zsz j Ω(N) Ω (N) Si z Sj z Ω (N) = T r [ G i 1 ZG j i 1 ZG N j] (.36) Z Z αβ;γδ = g i,αγ Sz i g i,βδ (.37) 4 4 (.34) (.36) G 1 G Z G = , Z = (.38) * 14 G 3-1(3 ) VBS N S z i S z j = 4 3 ( 1 3) j i (.39) VBS S x, S y ξ = 1 log 3 AKLT VBS Haldane.3.3 VBS (quantum disordered state) S = 1 S = 1 (string order) {S z i i = 1,..., N} ( 1 i, 0 i, 1 i ) (.31) VB g i 0 i g i,. = 1 i, g i,. = 1 i Ψ (N) i.αβ S z S z = 0 S z = 1 S z = 1 *13 C C.1.7 *14 C C.1.8
27 .3 Haldane 15 exp(iπs z ) S z = ±1-1 S z = 0 1 exp(iπ j 1 l=i+1 Sz l ) i j (i j) Sz = ±1 VBS Si z exp(iπ j 1 l=i+1 Sz l )Sz j ( 0) O z (i, j) = S z i exp(iπ j 1 l=i+1 S z l )S z j (.40) exp(iπ j 1 l=i+1 Sz l ) O z (i, j) = Ω (N) S z i exp(iπ j 1 l=i+1 P P αβ;γδ = g i,αγ exp(iπsz i )g i,βδ P = S z l )S z j Ω (N) (.41) (.4) VBS i j O z (i, j) = * 15 y S = 1 lim i j O α (i, j) VBS 84% VBS (valence bond).3.4 [ S= 1 ] (.8) VBS S = 1 S = 1 1 VBS S = 1 S = 1 S z S z i Ψ (N) αβ Sz i Ψ (N) αβ = (Gi 1 ZG N i )ᾱᾱ;ββ (.43) =, = Ψ (N) αβ Sz i Sz i αβ N 1 β Si z αβ = ɛ α ( 1 3 )i (.44) * 16 ɛ = 1, ɛ = 1 Si z αβ i Si z αβ = 1 ɛ α (.45) i=1 1 4 *15 C C.1.10 *16 C C.1.11
28 16 Kennedy S = 1 3 N N exp( N ξ ) 0 S = 1 1 VBS 1 Haldane Haldane Haldane gap.3.5 S > 1 S = 1 Haldane S S > 1 VBS VBS S S S = 1 S = 1 S S VBS S S = 1 S = 1 S = 1 S AKLT VBS VBS S z i exp(iθ j 1 l=i+1 Sz l )Sz j i j θ = π S S S= VBS 1 3 VBS S z VBS.3 S = 3 S = VBS S> 1 Haldane
29 .4 S=1 Bilinear-Biquadratic 17 (a) (b).3 VBS (a) (b).4 S=1 Bilinear-Biquadratic Haldane VBS VBS AKLT AKLT 1 1 S=1 Bilinear-Biquadratic.4.1 S=1 Bilinear-Biquadratic S=1 Bilinear-Biquadratic * 17 ( S=1 BLBQ ) Bilinear( ) Biquadratic( ) 1 H = S i S i+1a + α(s i S i+1 ) B (.46) [8] A Bilinear( ) B Biquadratic( ) 1 S=1 BLBQ 3 Na 87 Rb [9][30] 1 LiVGe O 6 [31] 1 α = 1 3 AKLT ( AKLT ) 1 < α < 1 Haldane α = 1 Takhtajan-Babujian(TB) [3][33][34] α = 1 Uimin-Lai-Sutherland(ULS) [35][36][37] α < 1 (gapped)dimer [38][39][8] α > 1 (gapless)trimer [40] (.4 ) Trimer Haldane Trimer c = BKT Haldane AKLT *17 1
30 18 Dimer Haldane C IC Trimer TB AKLT ULS α.4 S=1 BLBQ (α = 1 3 ) 1 < α < 1 3 (Commensurate)* < α < 1 (Incommensurate) * 19 [41][4] *18.4 C *19.4 IC
31 19 3 S=1 Bilinear-Biquadratic S=1 Bilinear-Biquadratic ( S=1 BLBQ ) S=1 Bilinear-Biquadratic S=1 BLBQ 1 [43] S=1 BLBQ AKLT VBS p 0.34 p 0.3 VBS H p = i h i,i+1,i+ (3.1) h i,i+1,i+ = 1 (S i S i+1 + S i+1 S i+ ) + 1 4p [(S i S i+1 ) + (S i+1 S i+ ) ] 6 +p[s i S i+ (S i S i+ ) ] (3.) S=1 Bilinear-Biquadratic H = S i S i+1 + α(s i S i+1 ) (3.3) AKLT (α = 1 3 ) VBS ( ) α α = VBS [43] S=1 BLBQ
32 0 3 S=1 Bilinear-Biquadratic 3. VBS S= [43] (AKLT )VBS S=1 1 (S=1 ) ( S=1 S=1 BLBQ ) [43] 1 ( 1) VBS ( ) 1 VBS ( 3) 3.. [ 1 ] 1 S=1 1 H J1,,J N,K 1,,K N H J1,,J N,K 1,,K N = 1 N N 1 [J k S i S i+k + K k (S i S i+k ) ] (3.4) i=1 k=1 ( S i i S=1 N ) J N m = J m K N m = K m S=1 1 H Jk,K k VBS VBS S = 1 i S = 1 i-1 i VBS G = Sym[ m ν m,m+1 ] (3.5) ( Sym (3 ) m ) ν i,j ( i j i j )/ *1 i j S = 1 1 * α i,j = Sym[ ν i,j ν i 1,j+1 m i 1,j ν m,m+1 ] (3.6) β i,j = Sym[ ν i 4,j+3 ν i 3,j+ ν j+1,i ν j,i 1 m i,i 1,j,j+1 ν m,m+1 ] (3.7) ( i < j ) i 1, M 1 ; i, M α = M 1 i1 1 M 1 i1 M i M i+1 m i1 1,i 1,i,i +1 0 m, (3.8) i 1, M 1 ; i, M β = M 1 i1 M 1 i1 1 M 1 i1 M i M i+1 M i+ m i1,i 1 1,i 1,i,i +1,i + 0 m (3.9) (i 1 < i ) M i i (S=1 )1 S z i M i = M M i α i 1, +1; i, 1 G, β i 1, +1; i, 1 G, α i 1, +1; i, 1 β i,j, β i 1, +1; i, 1 α i,j *1 ( i j i j )/ *.3. [43]
33 3. VBS S=1 1 *3 α i 1, +1; i, 1 α i,j, β i 1, +1; i, 1 β i,j i = i 1 j = i *4 VBS S=1 1 H Jk,K k β 1, +1; 1 + m, 1 H Jk,K k G = 0 (3.10) α 1, +1; 1 + m, 1 H Jk,K k G = 0 (3.11) *5 S i S i+1 G = 3 G + 1 α i,i+1, (3.13) (S i S i+1 ) G = 5 G 3 α i,i+1, (3.14) S i S i+ G = 1 [ G α i+1,i+ α i,i+1 + α i,i+ ], (3.15) (S i S i+ ) G = 1 [ G + α i+1,i+ + α i,i+1 + α i,i+ ], (3.16) S i S i+l G = 1 [ α i+1,i+l 1 α i+1,i+l α i,i+l 1 + α i,i+l ], (3.17) (S i S i+l ) G = 1 [ α i+1,i+l 1 + α i+1,i+l + α i,i+l 1 + α i,i+l ] + β i+1,i+l 1, (3.18) (l 3 ) (3.11) (3.18) l 3 K l = 0 VBS (bilinear) (J 1 J, J N ) (biquadratic) (K 1 K ) ( (J 1 J, J N K 1 K ) S=1 1 VBS S=1 H J1,,J N,K 1,K ) m 3 (3.11) J l + J l+ = J l+1 (l 3) (3.19) (3.19) J 3 = J 4 = m = 1 m = (3.11) K = J + J 3 (3.0) K 1 = (J 1 4J + 3J 3 )/3 (3.1) J 1 J J 3 K 1 K J l (l 4) J 1 J J 3 ( VBS S=1 H J1,,J N,K 1,K 3 J 1 J J 3 ( ) H J1,,J N,K 1,K H J1,J,J 3 ) *3 C C..1 *4 C C.. *5 G ( (3.4)) H G = G G β 1, +1; 1 + m, 1 H G = G β 1, +1; 1 + m, 1 G = 0 (3.1)
34 3 S=1 Bilinear-Biquadratic 3..3 VBS S=1 H J1,J,J 3 H J1,J,J 3 = J H 3 + (J 1 J 3 ) H p (3.) H = S i S j (3.3) (i,j) H p = i h i,i+1,i+ (3.4) h i,i+1,i+ = 1 (S i S i+1 + S i+1 S i+ ) + 1 4p [(S i S i+1 ) + (S i+1 S i+ ) ] 6 +p[s i S i+ (S i S i+ ) ] (3.5) ( p = (J J 3 ))/(J 1 J 3 ) (i,j) i j( i) ) 1 H VBS 1 1 H p VBS [ (+10p) 3 ]N H p VBS 3..4 H p VBS [ 3] 31 H p VBS [43] 16 KOBPACK *6 VBS ( ) p down 0.34 p up 0.3 (3.6) Lange 1 4 p 1 4 [44] ( Lange p = 1 4 ) 3..5 S=1 Bilinear-Biquadratic [43] S=1 BLBQ 3..3 VBS S=1 H J1,J,J 3 1 H p S=1 BLBQ (1 H p ) 3 S=1 BLBQ *6 KOBPACK A
35 3. VBS S=1 3 -p down p up 3.1 [43] N VBS p p up p down 0.4 -Pdown Pup -Pdown,Pup /N 3. KOBPACK 3.1
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37 5 4 Haldane 1 (AKLT ) (VBS ) 3 (VBS ) S=1 BLBQ ( (3.1) (3.)) S=1 Bilinear-Biquadratic ( S=1 BLBQ ) Haldane S=1 BlBQ λ ( ) λ = kλ k k [45][46] 4.1. S=1 Bilinear-Biquadratic S=1 BLBQ Haldane S=1 BLBQ Haldane (n 1) (Commensurate region) S0S z n z = ( 1) n n 1 exp( n /ξ(α)) (4.1) (Incommensurate region) S0S z n z = ( 1) n cos(q IC (α)n) n 1 exp( n /ξ(α)) (4.) ξ : [47] ( )
38 6 4 S0S z n z 0 S0 z n Sz n n *1 n * S=1 BLBQ ( 1 < α < 1 3 ) ( 1 3 < α < 1) 1 AKLT S0S z n z = ( 1) n exp( n /ξ(α)) (4.3) (disordered point) Haldane ( 4.1 ) α ξ(α) q IC (α) 4.1 [8] (α = 1 3 ) *3 4.1 Haldane (Commensurate region) ( 1 < α < 1 3 ) (Disordered point) (α = 1 3 ) (Incommensurate region) ( 1 3 < α < 1) S z 0S z n = ( 1) n n 1 exp( n /ξ(α)) S z 0S z n = ( 1) n exp( n /ξ(α)) S z 0S z n = ( 1) n cos(q IC (α)n) n 1 exp( n /ξ(α)) 4.1 (a) ξ(α) (b) q IC(α) *1 (4.1) n 1 exp( n /ξ(α)) n 1 S z n = * (4.) cos(q IC (α)n) n ( ) *3 (.39)
39 4. S=1 Bilinear-Biquadratic 7 4. S=1 Bilinear-Biquadratic 4..1 ( 1 3 < α < 1) q IC(α) [8] (open boundary condition : OBC) 3 (triplet) 1 (singlet) ( ST ) 3 ( triplet ) 1 ( singlet ) ST triplet singlet (4.4) (OBC) S=1 BLBQ N H = h i (4.5) i=1 h i = S i S i+1 + α(s i S i+1 ) (4.6) h i N h i L N N = L 1 (4.7) * 4 S=1 BLBQ triplet singlet N α triplet (N, α) singlet (N, α) N α ST (N, α) 4.. S=1 Bilinear-Biquadratic ST (N, α) S=1 BLBQ α N N α ( 4. ) n α α n (N) q IC (α) N α n (N) q IC (α n (N)) = πn N (4.9) [8][47][48] S=1 BLBQ (4.9) 4.3 ( L ) 4.1 q IC (α) *4 L S=1 BLBQ H = ˆS 1 S + α(s 1 S ) + + ˆS L 1 S L + α(s L 1 S L ) L 1 X = h i (4.8) N = L 1 i=1
40 8 4 1 AKLT ( VBS ) ST (N, α = 1 3 ) N.3.4 (OBC) VBS S=1 S = 1 S = 1 VBS 4 1 ( singlet ) 3 ( triplet ) AKLT (α = 1 3 ) 0 q IC (α 0 (N) = 1 3 ) = 0 π N = 0 (4.10) ( 4.3 L AKLT q IC (α(n)) = 0 ) 4.3 S=1 Bilinear-Biquadratic S=1 BLBQ q IC (α) S=1 BLBQ 4. L α ST [8] L=7 L=8 L=9 L=10 L=11 L=1 L=14 q ic AKLT α 4.3 ( 1 3 < α < 1) q IC(α)
41 4.3 S=1 Bilinear-Biquadratic S=1 Bilinear-Biquadratic S=1 BLBQ S=1 BLBQ AKLT VBS p 0.34 p 0.3 VBS VBS 4.1 S=1 BLBQ α VBS AKLT AKLT 1 S=1 BLBQ VBS S=1 BLBQ α 1 (AKLT ) S=1 BLBQ H p,β p S=1 BLBQ ( (3.1) (3.)) β β i 1 (S i S i+1 + S i+1 S i+ ) 1 H p,β H p,β = i [ h i,i+1,i+ + 1 ] β (S i S i+1 + S i+1 S i+ ) (4.11) = i [ 1 (1 + β)(s i S i+1 + S i+1 S i+ ) + 1 4p [(S i S i+1 ) + (S i+1 S i+ ) ] 6 +p[s i S i+ (S i S i+ ) ] ] (4.1) 1 H p,β p β H p,β S=1 BLBQ (A) S=1 BLBQ p 0.34 p 0.3 VBS 1 (A ) S=1 BLBQ p = 0 AKLT
42 30 4 *5 1 H p,β (B) 1 H p,β β (p = 0 ) S=1 BLBQ *6 S=1 BLBQ *7 (C) 1 H p,β β (p = 0 ) 1 < β 0 α AKLT ( 1 3 α) S=1 BLBQ β S=1 BLBQ α AKLT (α = 1 3 ) * *5 C C.3.1 *6 C C.3. *7.3. AKLT H AKLT = X i 1 S i S i (S i S i+1 ) (4.13).4 AKLT H AKLT = X i S i S i (S i S i+1 ) (4.14) *8 C C.3.3
43 4.3 S=1 Bilinear-Biquadratic 31 S=1 BLBQ Dimer Haldane C IC Trimer α β 0-1 p (A) H p,0 VBS (A ) p = 0 AKLT (B) β H 0,β S=1 BLBQ (C) β 1 < β 0 S=1 BLBQ α AKLT (C) H p,β β 0-1 p H p,β
44 H p,β (4.9) 4.6 KOBPACK *9 p β ( L n q(β n (N), p) = πn N ) p β q IC (β) ( q(β n (N), p) ) [ ] VBS S=1 BLBQ VBS 0.34 p 0.3 p p=0 S=1 BLBQ ( ) ( 4.1) ( ) S=1 BLBQ ( 4.1) β 0.6 ( (4.9)) S=1 BLBQ ( (4.9)) Haldane S=1 BLBQ ( 4.1) Haldane H p,β (B) (C) S=1 BLBQ Uimin-Lai-Sutherland(ULS) (α = 1) (Uimin-Lai-Sutherland ) H = S i S i+1 + (S i S i+1 ) (4.15) p = 0 β = 3 * 10 1 H p,β Haldane ( ) 1 H p,β Haldane ( ) ULS Unknown II Unknown I ( 4.15 ) Haldane ( ) 4.. ST (N, α) (N α ) ST (N, α) *9 KOBPACK A *10 (4.1) p = 0 β = 3 H 0, = X " # (1 3 3 ) S i S i (1 i 3 ) (S i S i+1 ) = X 1 ˆSi S i+1 + (S i S i+1 ) 3 i (4.16)
45 4.3 S=1 Bilinear-Biquadratic 33 q(β n (N), p) Unknown I II ST (N, α) ( 4.6 Unknown I II q(β n (N), p) ) ( ) 6 7 Unknown II, 8 Unknown I
46 β L= 4 0 0/ π L= 4 1 1/ π L= 5 0 0/ 3π L= 5 1 1/ 3π L= 5 / 3π L= 7 0 0/ 5π L= 7 1 1/ 5π L= 7 / 5π L= 7 3 3/ 5π L= 7 4 4/ 5π L= 7 5 5/ 5π L= 7 6 6/ 5π L= 7 7 7/ 5π L= 7 8 8/ 5π L= 7 9 9/ 5π L= 8 0 0/ 6π L= 8 1 1/ 6π L= 8 / 6π L= 8 3 3/ 6π L= 9 0 0/ 7π L= 9 1 1/ 7π L= 9 / 7π L= 9 3 3/ 7π L= 9 4 4/ 7π L= 9 5 5/ 7π L= 9 6 6/ 7π L= 9 7 7/ 7π L= 9 8 8/ 7π L= 9 9 9/ 7π L= / 7π L= / 7π L= 9 1 1/ 7π L= / 7π L= / 7π L= / 7π L= / 7π L= / 7π L= / 7π L= / 7π L= 9 0 0/ 7π L=10 0 0/ 8π L=10 1 1/ 8π L=10 / 8π L=10 3 3/ 8π L=10 4 4/ 8π L=11 0 0/ 9π L=11 1 1/ 9π L=11 / 9π L=11 3 3/ 9π L=11 4 4/ 9π L=11 5 5/ 9π L=11 6 6/ 9π L=11 7 7/ 9π L=11 8 8/ 9π L=1 0 0/10π L=1 1 1/10π L=1 /10π L=1 3 3/10π L=1 4 4/10π L=1 5 5/10π L=13 0 0/11π L=13 1 1/11π L=13 /11π L=13 3 3/11π L=13 4 4/11π L=13 5 5/11π L=13 6 6/11π L=13 7 7/11π L=13 8 8/11π L=13 9 9/11π L= /11π L= /11π L=13 1 1/11π L= /11π L= /11π L= /11π L= /11π L= /11π L= /11π L= /11π L=13 0 0/11π L=13 1 1/11π L=13 /11π L=13 3 3/11π L=13 4 4/11π L=13 5 5/11π L=13 6 6/11π P 4.6 S=1 BLBQ
47 4.3 S=1 Bilinear-Biquadratic 35 q(β, p) P= L= 4 0 0/ π L= 5 0 0/ 3π L= 7 0 0/ 5π L= 8 0 0/ 6π L= 9 0 0/ 7π L=10 0 0/ 8π L=11 0 0/ 9π L=1 0 0/10π L=13 0 0/11π L=13 1 1/11π L=1 1 1/10π L=11 1 1/ 9π L=10 1 1/ 8π L= 9 1 1/ 7π L= 8 1 1/ 6π L=13 /11π L= 7 1 1/ 5π L=1 /10π L=11 / 9π L=10 / 8π L=13 3 3/11π L= 9 / 7π L=1 3 3/10π L= 5 1 1/ 3π L= 8 / 6π L=11 3 3/ 9π L=10 3 3/ 8π L=1 4 4/10π L= 9 3 3/ 7π L= 4 1 1/ π L= 5 / 3π β 4.7 P=0 ( 4.6 ) q IC(β) q(β, p) P= L= 4 0 0/ π L= 5 0 0/ 3π L= 7 0 0/ 5π L= 8 0 0/ 6π L= 9 0 0/ 7π L=10 0 0/ 8π L=11 0 0/ 9π L=1 0 0/10π L=13 0 0/11π L=13 1 1/11π L=1 1 1/10π L=11 1 1/ 9π L=10 1 1/ 8π L= 9 1 1/ 7π L= 8 1 1/ 6π L=13 /11π L= 7 1 1/ 5π L=1 /10π L=11 / 9π L=10 / 8π L=13 3 3/11π L= 9 / 7π L=1 3 3/10π L= 5 1 1/ 3π L= 8 / 6π L=11 3 3/ 9π L=13 4 4/11π L=10 3 3/ 8π L= 7 / 5π L=1 4 4/10π L= 9 3 3/ 7π L=11 4 4/ 9π L=13 5 5/11π L= 4 1 1/ π L= 8 3 3/ 6π L=10 4 4/ 8π L=1 5 5/10π L=11 5 5/ 9π L= 7 3 3/ 5π β 4.8 P=0.06 ( 4.6 ) q IC (β)
48 36 4 q(β, p) P= L= 4 0 0/ π L= 5 0 0/ 3π L= 7 0 0/ 5π L= 8 0 0/ 6π L= 9 0 0/ 7π L=10 0 0/ 8π L=11 0 0/ 9π L=1 0 0/10π L=13 0 0/11π L=13 1 1/11π L=1 1 1/10π L=11 1 1/ 9π L=10 1 1/ 8π L= 9 1 1/ 7π L= 8 1 1/ 6π L=13 /11π L= 7 1 1/ 5π L=1 /10π L=11 / 9π L=10 / 8π L=13 3 3/11π L= 9 / 7π L=1 3 3/10π L= 5 1 1/ 3π L= 8 / 6π L=11 3 3/ 9π L=13 4 4/11π L=10 3 3/ 8π L= 7 / 5π L=1 4 4/10π L= 9 3 3/ 7π L=11 4 4/ 9π L=13 5 5/11π L= 4 1 1/ π L= 8 3 3/ 6π L=10 4 4/ 8π L=1 5 5/10π L=11 5 5/ 9π L= 7 3 3/ 5π β 4.9 P=0.15 ( 4.6 ) q IC(β) q(β, p) P= L= 4 0 0/ π L= 5 0 0/ 3π L= 7 0 0/ 5π L= 8 0 0/ 6π L= 9 0 0/ 7π L=10 0 0/ 8π L=11 0 0/ 9π L=1 0 0/10π L=13 0 0/11π L=13 1 1/11π L=1 1 1/10π L=11 1 1/ 9π L=10 1 1/ 8π L= 9 1 1/ 7π L= 8 1 1/ 6π L=13 /11π L= 7 1 1/ 5π L=1 /10π L=11 / 9π L=10 / 8π L=13 3 3/11π L= 9 / 7π L=1 3 3/10π L= 5 1 1/ 3π L= 8 / 6π L=11 3 3/ 9π L=13 4 4/11π L=10 3 3/ 8π L= 7 / 5π L=1 4 4/10π L= 9 3 3/ 7π L=11 4 4/ 9π L=13 5 5/11π L= 4 1 1/ π L= 8 3 3/ 6π L=10 4 4/ 8π L=1 5 5/10π L=13 6 6/11π L=11 5 5/ 9π L= 9 4 4/ 7π L= 7 3 3/ 5π L=13 7 7/11π L=11 6 6/ 9π L= 9 5 5/ 7π L=13 8 8/11π L= 7 4 4/ 5π L=13 9 9/11π L= 9 6 6/ 7π L= /11π L= 9 7 7/ 7π L= /11π L=13 1 1/11π L= 9 8 8/ 7π L= /11π L= /11π L= 9 9 9/ 7π L= /11π L= / 7π L= /11π L= /11π L= / 7π L= /11π L= 9 1 1/ 7π L= /11π L=13 0 0/11π L= / 7π L=13 1 1/11π L= / 7π L=13 /11π L=13 3 3/11π L= / 7π L=13 4 4/11π L= / 7π L= / 7π L= / 7π L= / 7π L= 9 0 0/ 7π β 4.10 P=0.1 ( 4.6 ) q IC (β)
49 4.3 S=1 Bilinear-Biquadratic 37 q(β, p) P= L= 7 0 0/ 5π L= 9 0 0/ 7π L=13 0 0/11π L=13 1 1/11π L= 9 1 1/ 7π L=13 /11π L= 7 1 1/ 5π L=13 3 3/11π L= 9 / 7π L=13 4 4/11π L= 7 / 5π L= 9 3 3/ 7π L=13 5 5/11π L=13 6 6/11π L= 9 4 4/ 7π L=13 7 7/11π L= 9 5 5/ 7π L=13 8 8/11π L=13 9 9/11π L= 9 6 6/ 7π L= 9 7 7/ 7π β 4.11 P=0.33 ( 4.6 ) q IC(β) q(β, p) P= L= 4 0 0/ π L= 5 0 0/ 3π L= 7 0 0/ 5π L= 8 0 0/ 6π L= 9 0 0/ 7π L=10 0 0/ 8π L=11 0 0/ 9π L=1 0 0/10π L=13 0 0/11π L=13 1 1/11π L=1 1 1/10π L=11 1 1/ 9π L=10 1 1/ 8π L= 9 1 1/ 7π L= 8 1 1/ 6π L=13 /11π L= 7 1 1/ 5π L=1 /10π L=11 / 9π L=10 / 8π L= 9 / 7π L=1 3 3/10π L= 5 1 1/ 3π L= 8 / 6π L=11 3 3/ 9π L=1 4 4/10π L= 9 3 3/ 7π L= 4 1 1/ π L= 5 / 3π β 4.1 P=-0.06 ( 4.6 ) q IC (β)
50 38 4 q IC (β) P= N= 4 0 0/ π N= 5 0 0/ 3π N= 7 0 0/ 5π N= 8 0 0/ 6π N= 9 0 0/ 7π N=10 0 0/ 8π N=11 0 0/ 9π N=1 0 0/10π N=13 0 0/11π N=13 1 1/11π N=1 1 1/10π N=11 1 1/ 9π N=10 1 1/ 8π N= 9 1 1/ 7π N= 8 1 1/ 6π N=13 /11π N= 7 1 1/ 5π N=1 /10π N=11 / 9π N=10 / 8π N= 9 / 7π N=1 3 3/10π N= 5 1 1/ 3π N= 8 / 6π N= 4 1 1/ π N= 5 / 3π β 4.13 P=-0.15 ( 4.6 ) q IC(β) q(β, p) P= L= 4 0 0/ π L= 5 0 0/ 3π L= 7 0 0/ 5π L= 8 0 0/ 6π L= 9 0 0/ 7π L=10 0 0/ 8π L=11 0 0/ 9π L=1 0 0/10π L=13 0 0/11π L=13 1 1/11π L=1 1 1/10π L=11 1 1/ 9π L= 9 1 1/ 7π L= 8 1 1/ 6π L=13 /11π L= 7 1 1/ 5π L=1 /10π L=11 / 9π L= 9 / 7π L= 5 1 1/ 3π L= 9 3 3/ 7π L= 4 1 1/ π L= 5 / 3π β 4.14 P=-0.1 ( 4.6 ) q IC (β)
51 4.3 S=1 Bilinear-Biquadratic Unknwon
52
53 H p,β ( ) [1][49] 1 ( ) 5.1 S H = x,x J( x x )S(x) S(x ) (5.1) J( x x ) S x J ( ) x,x x = x [S i, S j ] = iɛ ijk S k (i, j, k = 1,, 3) (5.) ɛ ijk : * 1 (5.3) [S 1 (x), S (x )] = iδ x,x S 3 (x) ( ) (5.4) x x, s, m S (x) x, s, m = s(s + 1) x, s, m (5.5) S z (x) x, s, m = m x, s, m (5.6) (m = s, s 1,..., s) *1 ɛ ijk (.1)
54 ( (5.1)) m = s x, s, m S ± S x ± is y (5.7) [ S z (x), S ± (x ) ] = ±δ x,x S ± (x) (5.8) [ S + (x), S (x ) ] = δ x,x S z (x) (5.9) S + (x) x, s, s = 0 (5.10) S z (x) x, s, s = s x, s, s (5.11) * Φ 0 Π x x, s, s (5.1) H = [ J( x x ) S z (x)s z (x ) + 1 ( S + (x)s (x ) + S (x)s + (x ) )] (5.13) x,x H Φ 0 (5.10) (5.11) HΦ 0 = 0 Φ 0 (5.14) 0 = x,x J( x x )m (5.15) Φ 0 H 5.3 [ H, S (x) ] = J( x x ) ( S (x)s z (x ) S (x )S z (x) ) (5.16) x *3 (5.8) (5.9) x S z (x) s 1 Φ(x) Φ(x) S (x)φ 0 (5.17) HΦ(x) = H S (x)φ 0 = S (x) HΦ 0 + [H, S (x)] Φ 0 = 0 S (x)φ 0 + s J( x x ) ( S (x) S (x ) ) Φ 0 x = 0 Φ(x) + s x J( x x )Φ(x) s x J( x x )Φ(x ) (5.18) * (5.10) B (B.4) *3 C C.4.1
55 *4 Φ(x) H Φ(x) 1 Φ 1 x a(x)φ(x) (5.19) HΦ 1 = 0 Φ 1 + s x J( x )Φ 1 s x,x J( x x )a(x)φ(x ) (5.0) a(x) = 1 Ns e i k x (N = N x N y N z ) (5.1) (N x, N y, N z ) Φ 1 Φ(k) Φ(k) = 1 e i k x Φ(x) = 1 e i k x S (x)φ 0 (5.) Ns Ns x x HΦ(k) = 0 Φ(k) + s J( x )Φ(k) s J( x x 1 ) e i k x Φ(x ) x x,x Ns ( = 0 + s ) J( x ) Φ(k) s J( x x 1 ) e i kx Φ(x ) x x,x Ns ( = 0 + s ) J( x ) Φ(k) s J( x )e i k x Φ(k) (5.3) x x *5 ɛ(k) = s x J( x )(1 e i k x ) (5.5) (H 0 )Φ(k) = ɛ(k)φ(k) (5.6) Φ(k) H k k = k x b x + k y b y + k z b z (b x, b y, b z ) (5.7) *4 P x J( x x )Φ(x) = P x J( x )Φ(x) P x x Φ(x) P x x = x J( x x ) x x *5 P x,x J( x x 1 ) e i kx Φ(x ) = P Ns x J( x )e i kx Φ(k) X J( x x 1 ) e i k x Φ(x ) = X J( x x 1 ) e i k (x x ) e i k x Φ(x ) Ns x,x Ns x,x X x x,x = X! X J( x x 1 ) e i k (x x ) e i k x Φ(x ) x x Ns x 1 Ns e i k x Φ(x ) = X x X! 1 Ns e i k x Φ(x ) J( x )e i k x x X x = 0 x = X x J( x )e i k x Φ(k) (5.4)
56 44 5 k i = /N i (i = x, y, z) ɛ(k) k k + K Φ(k) Φ(k ) = δ k,k (5.8) *6 (5.6) k ɛ(k) ɛ(k) ( ) * H ( (5.1)) J > 0 Φ 0 ( *8 )Φ(k) Φ 0 Φ(k) Φ(k) 0 Φ(k) = 1 Ns x ei k x Φ(x) 0 + ɛ(k) ( (5.15)) ( (5.)) ( (5.6)) 5.4 S=1 Bilinear-Biquadratic S=1 BLBQ S=1 BLBQ β β i S i S i+1 1 H p,β = i = i S i S i p (S i S i+1 ) + p[s i S i+ (S i S i+ ) ] + β S i S i+1 3 (1 + β)s i S i p (S i S i+1 ) + p[s i S i+ (S i S i+ ) ] 3 (5.30) *6 C C.4. *7 ɛ(k) [49]p93 *8 Φ(k) J > 0 Φ 0 Φ(k) ( 0 + ɛ(k)) 0 = ɛ(k) = s X J( x )(1 e i k x ) (5.9) x J > 0 ɛ(k) > 0 Φ(k) Φ 0 Φ(k)
57 5.4 S=1 Bilinear-Biquadratic x S(x) a i x = n x a x + n y a y + n z a z (n x, n y, n z ) (5.31) e i x = n x e x *9 n x = i S(x = i) = S i (5.3) (i ie x ) ( ) 1 Φ 0 = Π i i, s, s (5.33) Φ(k) = 1 e i k i Φ i = 1 e i k i S i Φ 0 (5.34) Ns Ns i Φ(k m ) = 1 e i kmi Φ i = 1 e i kmi S i Φ 0 (5.35) Ns Ns i Φ(x = i) = Φ i k = m/n b x (m ) i = ie x k i = m/n b x ie x = π i m/n i i = πm N i = k m i (5.36) (5.30) 1 (5.30) 1 [ N i=1 S i S i+1, S j ] [ N ] S i S i+1, S j = Sj z S j+1 + S j 1 Sz j S j Sz j+1 Sj 1S z j (5.37) i=1 *9 1 1
58 46 5 * 10 N i=1 S i S i+1 Φ(k m ) N N S i S i+1 Φ(k m ) = S i S i+1 1 e i kmj S j Φ 0 Ns i=1 i=1 = 1 Ns j e i k mj j ( N ) S i S i+1 S j Φ 0 i=1 ( = 1 N ) e i kmj S j S i S i+1 Ns j i=1 Φ 0 [ + 1 N ] e i kmj S i S i+1, S j Ns j i=1 ( ) N Φ 0 0 ( S i S i+1 )Φ 0 = 0 Φ 0 1 = 0 e i kmj S j Φ Ns Ns Φ 0 i=1 e i k mj j j i=1 [ N ] S i S i+1, S j = 0 Φ(k m ) + [cos(k m ) 1] Φ(k m ) = ( 0 + [cos(k m ) 1]) Φ(k m ) (5.38) * 11 Φ 0 (5.30) 3 3 [ N i=1 S i S i+, S j ] [ N ] S i S i+, S j = Sj z S j+ + S j Sz j S j Sz j+ Sj S z j (5.39) i=1 * 1 N i=1 S i S i+ Φ(k m ) N N S i S i+ Φ(k m ) = S i S i+ 1 e i kmj S j Φ 0 Ns i=1 i=1 = 1 Ns j j e i k mj j ( N ) S i S i+ S j Φ 0 i=1 ( = 1 N ) e i kmj S j S i S i+ Ns i=1 Φ 0 [ + 1 N ] e i kmj S i S i+, S j Ns j i=1 ( ) N Φ 0 0 ( S i S i+ )Φ 0 = 0 Φ 0 Φ 0 i=1 [ = 0 Φ(k m ) + 1 N ] e i kmj S i S i+, S j Ns j i=1 = 0 Φ(k m ) + [cos(k m ) 1] Φ(k m ) = ( 0 + [cos(k m ) 1]) Φ(k m ) (5.40) *10 C C.4.3 h *11 1 PN i Ns Pj ei k mj i=1 S i S i+1, S j Φ 0 = [cos(k m ) 1] Φ(k m ) C C.4.4 *1 C C.4.3 Φ 0
59 5.4 S=1 Bilinear-Biquadratic 47 * 13 (5.30) 4 (5.30) 1 3 ( )Φ 0 1 Φ(k m ) 4 B (B.11) (S i S i+1 ) (S i S i+ ) Φ 0 Φ(k m ) (S i S i+1 ) (S i S i+1 ) Φ 0 = (S i S i+1 ) Π j j, s, s s = 1 = (S i S i+1 ) Π j j, 1, 1 = (S i S i+1 ) i, 1, 1 i + 1, 1, 1 Π j i,i+1 j, 1, 1 B (B.11) * 14 = i, 1, 1 i + 1, 1, 1 Π j i,i+1 j, 1, 1 = Φ 0 (5.41) (S i S i+1 ) Φ(k m ) = (S i S i+1 ) 1 e i kmi S l Π j j, 1, 1 Ns = 1 e i kmi (S i S i+1 ) S l Π j j, 1, 1 Ns l l ( ) = 1 e i kmi (S i S i+1 ) S l i, 1, 1 i + 1, 1, 1 Π j i,i+1 j, 1, 1 Ns l l i, i + 1 (5.41), l = 1 ) (e i kmi (S i S i+1 ) S i + e i km(i+1) (S i S i+1 ) S i+1 i, 1, 1 i + 1, 1, 1 Π j i,i+1 j, 1, 1 Ns + 1 Ns l ı,i+1 e i k mi S l Π j j, 1, 1 = 1 ( e i k mi (S i S i+1 ) i, 1, 0 i + 1, 1, 1 Π j i,i+1 j, 1, 1 Ns ) +e i km(i+1) (S i S i+1 ) i, 1, 1 i + 1, 1, 0 Π j i,i+1 j, 1, Ns l ı,i+1 e i k mi Π j j, 1, 1 B (B.11) * 15 = 1 e i kmi S l Π j j, 1, 1 Ns l = Φ(k m ) (5.4) *13 1 P h PN i Ns j ei kmj i=1 S i S i+, S j Φ 0 = [cos(k m ) 1] Φ(k m ) C.4.5 *14 (S i S i+1 ) i, 1, 1 i + 1, 1, 1 = i, 1, 1 i + 1, 1, 1 B B. i, 1, 1 i + 1, 1, 1 = 1 i 1 i+1
60 48 5 Φ 0 1 Φ(k m ) Φ(k m ) ( ) (1 + β)s i S i+1 + ps i S i+ Φ(k m ) i = ((1 + β) ( 0 + [cos(k m ) 1]) + p ( 0 + [cos(k m ) 1])) Φ(k m ) ( ) = (1 + β + p) 0 + (1 + β) [cos(k m ) 1] + p [cos(k m ) 1] Φ(k m ) (5.44) Φ 0 ( ) (1 + β)s i S i+1 + ps i S i+ Φ 0 = (1 + β + p) 0 Φ 0 (5.45) i Φ 0 Φ(k m ) (1+β) [cos(k m ) 1]+p [cos(k m ) 1] 1 Φ(k m ) Φ 0 (1 + β) [cos(k m ) 1] + p [cos(k m ) 1] 0 (5.46) (5.46) (1 + β) 0 p (1 + β) 1 4 (5.47) * 16 β, p ( (5.47)) (k) = (1 + β) [cos(k m ) 1] + p [cos(k m ) 1] (5.48) (k) < 0 k *15 (S i S i+1 ) i, 1, 0 i + 1, 1, 1 = i, 1, 0 i + 1, 1, 1 (S i S i+1 ) i, 1, 1 i + 1, 1, 0 = i, 1, 1 i + 1, 1, 0 (5.43) B B. i, 1, 0 i + 1, 1, 1 = 0 i 1 i+1 i, 1, 1 i + 1, 1, 0 = 1 i 0 i+1 *16 C C.4.6
61 5.5 S=1 Bilinear-Biquadratic S=1 Bilinear-Biquadratic 1 H p,β ( (5.30)) Φ 0 Φ 0 Φ(k m ) Φ 0 Φ(k m ) (1 + β + p) 0 ((1 + β + p) 0 + (1 + β) [cos(k m ) 1] + p [cos(k m ) 1]) ( (5.45)) ( (5.44)) Φ 0 Φ(k m ) ( (5.46)) (1 + β) 0 p (1+β) 1 4 (5.47) β = 1, p 0 k = π β < 1, p = 1 1+β 4 (1 + β) k = ± arccos ( 4p ) ( 5.1 ) * 17 p p (β = 1 p = 0) β k = ± arccos ( 1+β 4p ) (β = 1 p = 0) β k = π β = 1 p = 0 1 k (k) = (1 + β) [cos(k m ) 1] + p [cos(k m ) 1] = 0 (5.49) ( C C.4.7 C.10 ) 1 H 0, 1 L (L: ) [50] 5.5 S=1 Bilinear-Biquadratic ( )Φ Φ(k m ) ( (5.46)) 1 H p,β Φ (Φ(k m )) 0 (Φ(k m )) 0 0 (5.50) *17 C C.4.7
62 50 5 KOBPACK * 18 L = 1, 15 Stotal z (= k Sz k ) l 0 (S z total = l) Φ 0 0 F (l) = 0 (S z total = l) 0 (5.51) z ( ) F (l) 0 (l) ( ) p = 1 4 (1 + β) L Φ = 0 (S z total = L) (5.5) ( L = 1 Φ (S z total = 1) ) F (0) F (1) F (L 1) F (L) F (l) (3 l L ) (1 + β) 0 p (1 + β) 1 4 Φ 0 Φ (Stotal z = l) z F = 0 (S total = 0) - z 0 (S total =1) z F = 0 (S total = 1) - z 0 (S total =1) F P β F P β 5. L=1 F (0) 5.3 L=1 F (1) *18 KOBPACK A
63 5.5 S=1 Bilinear-Biquadratic 51 z F = 0 (S total = 0) - z 0 (S total =1) z F = 0 (S total = 1) - z 0 (S total =1) P P β β F (0) F (1) z F = 0 (S total =10) - z 0 (S total =1) z F = 0 (S total =11) - z 0 (S total =1) F P β F P β 5.6 L=1 F (10) 5.7 L=1 F (11) z F = 0 (S total =10) - z 0 (S total =1) z F = 0 (S total =11) - z 0 (S total =1) P P β β F (10) F (11)
64
65 53 6 Uimin-Lai-Sutherland 4 p = 0 β = 3 ( )Uimin-Lai-Sutherland(ULS) 1 H p,β p = 0 β = 3 ULS p 0 S=1 Bilinear-Biquadratic ( S=1 BLBQ ) Haldane Trimer (.4 ) Uimin-Lai-Sutherland(ULS) *1 (ULS ) ULS 1 H p,β ULS p 0 ULS ULS 6.1 Uimin-Lai-Sutherland Uimin-Lai-Sutherland(ULS) Bethe Wess-Zumino-Witten(WZW) massless ( ) Uimin-Lai-Sutherland ULS I * q = ± 3π [51][5][53] II SU(3) [54][55] ULS SU(3) SU() 3 (triplet) 5 (quintet) SU(3) BKT [56] ULS I II 1 H p,β *1 1 H p,β p = 0 β = 3 1 H p,β ULS * 7..
66 54 6 Uimin-Lai-Sutherland 6.1. ULS I II? I II 3 3 (triplet) 5 (quintet) ( ) I q = ± 3π( 3 ) II SU(3) 3 (triplet) 5 (quintet) ULS SU(3) 6. Uimin-Lai-Sutherland SU(3) 6..1 ULS SU(3) S=1 H(θ) = L [ cos θ(sj S j+1 ) + sin θ(s j S j+1 ) ] (6.1) j=1 1 θ [θ [0,π)] θ = π/4 H(γ) = 1 L cos θ H(θ) = [ (Sj S j+1 ) + γ(s j S j+1 ) ] (6.) j=1 (γ = tan θ ) c j,α c j,α S j = 3 αβ=1 ( )(L x L y L z S=1 ) c j,α (L) αβc j,β (6.3) (L i ) αβ (L i ) γη = δ βγ δ αη δ αγ δ βη, (6.4) [ L i L j] [ αβ L i L j] γη = δ αβδ γη δ αγ δ βη (6.5) S j S j+1 = c j,α c j,βc j+1,β c j+1,α c j,α c j,βc j+1,α c j+1,β (6.6)
67 6. Uimin-Lai-Sutherland SU(3) 55 *3 ( ) 3 c j,α c j,α = 1 (6.7) α=1 *4 (S j S j+1 ) = 1 + c j,α c j,βc j+1,α c j+1,β (6.8) *5 (6.) (6.6) (6.8) H(γ) = = L [ (Sj S j+1 ) + γ(s j S j+1 ) ] j=1 L j=1 [ ] c j,α c j,βc j+1,β c j+1,α + (γ 1)c j,α c j,βc j+1,α c j+1,β (6.9) 6.. Uimin-Lai-Sutherland SU(3) (6.9) γ = 1 ULS H(γ = 1) = L j=1 [ ] c j,α c j,βc j+1,β c j+1,α (6.10) SU(3) 3 3 U c j,α (U) αβ c j,β (6.11) β ( (6.11)) H(1) = ] [[(U) αγ ] c j,γ (U) βηc j,η [(U) βγ ] c j+1,γ (U) αη c j+1,η j=1 (6.1) ] H(1) = [[(U) αγ ] (U) αη (U) βη [(U) βγ ] c j,γ c j,ηc j+1,γ c j+1,η ] [δ γη δ ηγ c j,γ c j,ηc j+1,γ c j+1,η = j=1 = j=1 [ ] c j,γ c j,ηc j+1,η c j+1,γ γ α η β = (6.10) (6.13) (U) αβ ] (U) αγ = (U) βα ] (U) αγ = δ βγ ULS SU(3) [54][55] *3 C C.5.1 *4 C C.5. *5 C C.5.3
68 56 6 Uimin-Lai-Sutherland KOBPACK *6 3 L= ( ) L= p cross,l (β) x 1 L y p cross (β) p cross (β) = a + b 1 L (6.14) a b (gnuplot fit ) a L = ( ) p cross, (β) β β = 0.66 β = ( β = 0.66( 0.68) p cross, (β) = ( ) ) ( 6.7 ) p 0 ULS I II ( ) ULS ( ) [56] ( ) P 0.4 L= ULS POINT / β L=9 triplet-quintet 6. L=1 triplet-quintet P 0.4 L= ULS POINT /3 β *6 KOBPACK A
69 [56] S=1 L [ H(θ) = cos θ(sj S j+1 ) + sin θ(s j S j+1 ) ] (6.15) j=1 θ = π 4 (ULS ) θ > π 4 massless ULS massless Haldane Berezinskii-Kosterlitz-Thouless(BKT) ( Fath Solyom [57] ) ULS SU(3)Wess-Zumino-Witten(WZW) ( Bethe ) ULS SU(3) WZW SU(3) (marginal operator) ( SU(3) 0.4 L= L= ULS POINT ULS POINT P 0 P / / β β 6.3 L=15 triplet-quintet 6.4 L=18 triplet-quintet P cross (β = -0.66) p cross (β = -0.66)=a + b*1/l a= /-.4111e-07( %) b= / e-05( %) P cross, (β)= /L P cross (β =-0.68) p cross (β =-0.68)=a + b*1/l a= / e-06( %) b= / ( %) P cross, (β)= /L 6.5 p cross (β = 0.66) = a + b 1 L 6.6 p cross (β = 0.68) = a + b 1 L
70 58 6 Uimin-Lai-Sutherland BKT ) (conformal field theory(cft)) BKT SU() ν > SU(ν) ν > BKT ν = massless 1 Ludwig Cardy[58][59] L (Ground-State(GS)) ε GS = ɛ πv 6L c(l) c(l) = c vir + d GS (ln L) 3 + O ( ) ln (ln L) (ln L) 4, 1 (6.16) (ln L) 4 c(l) ɛ v *7 x n (primary field) ε n = ε GS + πv L γ n(l) γ n (L) = x n + d n (ln L) + O ( ln (ln L) (ln L), 1 (ln L) ) (6.17) x n d n 3 (x n d n n ) 0.4 Fitting L= 9,1, ULS POINT P / β 6.7 L=9,1,15 fit *7 7..
71 [56] (6.16) (6.17) [56] (6.16)(6.17) 1, SU(3) Wess-Zumino-Witten ( SU(3) SU() ), BKT massless c = 3, 6. γ ( (6.15)) 6. S=1 (6.16)(6.17) SU(3) WZW (γ = 1) 3 SU(3) WZW (γ 1) 3 Bethe 3 c vir x q x t x s d GS d q d t d s H(γ) = 1 L cos θ H(θ) = [ (Sj S j+1 ) + γ(s j S j+1 ) ] (6.18) j=1 γ = tan θ x q d q x t d t x s d s 5 (quintet) 3 (triplet) 1 (singlet) [56] ULS c = BKT BKT ULS BKT BKT.1.3 S= 1 XY 1 XXZ 1 XXZ massless c = 1 nassless c = ULS BKT
72
73 H p,β p 0 Uimin-Lai- Sutherland(ULS) I II ( ) *1 ( ) q 1 H p,β 7.1 c = (CFT) ( (6.17)) 1 ln L ( (6.16)) 1 (ln L) { (T 0) 1 (T = 0) ( (ξ = )) 7.1. [massless(ξ = ) ] c 0 (L) = ɛ 0 L πvc 6L (7.1) *1 I II 6.1.1
74 6 7 L : c : v : ɛ 0 : 0 (L) : L L v v = L π (q = π L ) (7.) (q = π L ) (q = π L ) 0(q = π L ) 0(q = 0) (7.3) 0 (q = π L ) : q = π L 0 (q = 0) : q = 0 q ( ) ( ) * ( p ) [massive( 1 ξ 0) ] (7.1) (7.3) 0 (L) L = ɛ 0 c exp ( L ξ ) (7.4) (q = π L ) v (q + ξ ) L const (7.5) 7. c q 7..1 [60] (D ) ( ) H = [ J (S z i S z j ) + J (S + i S j + S i S + j )] D [ ] 1 3 S(S + 1) (Sz k) (7.6) <i,j> * k
75 7. 63 ( <i,j> 1 ) H H Stotal z (= k Sz k ) H Sz total H Stotal z ( Stotal z ) ( S z ) Stotal z S z *3 7.. [ ] n ( H n ) Bonner Fisher i i+1 ( ) T T n 1 πl i p, l = e n j (T ) j p (7.7) j=0 e i πl n *4 l 0 < l < n p ( p Si z ) T H *5 ( l *6 ) Bonner Fisher e i πl n q(= πl n πl n ) H T q(= πl n ) p H i (H i = i i ) q(= πl n ) H i, l (1 ) q (7.7) ( ) q = πl L π(n l) q = L *7 *3 S z total S z total H ( ) Sz total S z total *4 C C.6.1 *5 Stotal z T *6 C C.6. *7 A A A.5. C C.6.7
76 i n-i+1 ( ) U H T UT = T 1 U * 8 p ± 1 p ±, l = (1 ± U)( p, l + p, n l ), (7.8), l = i(1 ± U)( p, l p, n l ) (7.9) *9 p, l * 10 * 11 l 0 l n p i, l p + πl i, l q = L ( q = π(n l) L ) * p KOBPACK q KOBPACK (7.8) (7.9) p 1, l = 1 N ( p, l + p, n l ), (7.10) p, l = 1 N ( p, l p, n l ) (7.11) ( * 13 ) KOBPACK q KOBPACK 3 Lanczos A 7.3 [ ] [ ] *8 i i T i + 1 U n (i + 1) + 1 = n i i U n i + 1 T 1 n i UT = T 1 U *9 T + T 1 [ C C.6.6 ] *10 C C.6.4 *11 C C.6.5 *1 C C.6.7 *13 C C.6.8
77 7.3 [ ] 65 (7.1) ɛ L 3 L+3 0 (L) = ɛ 0 L πvc 6L 0 (L + 3) = ɛ 0 (L + 3) πvc 6(L + 3) (7.1) (7.13) * 14 0 (L) = ɛ 0 πvc L 6L (7.14) 0 (L + 3) πvc = ɛ 0 L + 3 6(L + 3) (7.15) (7.15) (7.14) 0 (L + 3) L + 3 0(L) L ( ɛ 0 ) ( ɛ 0 πvc ) 6L πvc = 6(L + 3) = πvc 6L πvc 6(L + 3) ( πv ) ( (L + 3) L ) = 6 L (L + 3) c ( πv ) ( ) 6L + 9 = 6 L (L + 3) c c ( ) ( 6 L (L + 3) ) ( 0 (L + 3) c eff = ) 0(L) πv 6L + 9 L + 3 L (7.16) (7.17) (7.1) (7.15) v v = 1 ( L π (q = π L ) + L + 3 ) π (q = π L + 3 ) (7.18) ɛ 0 c eff L L [c eff c] (7.17) c c eff (7.1) L ( L=18) c eff c c eff massless ( ) lim eff = c L massive ( ) lim eff = 0 L *14 1 H p,β ( 1 )
78 β p (7.17) p ULS (β ) - [56] Trimer c = H p,β (B) 1 H p,β β (p = 0 ) 1 < β 0 S=1 BLBQ (C) S=1 BLBQ 1 H p,β β β S=1 BLBQ α AKLT (α = 1 3 ) 1 H p,β ULS (p = 0 β = 3 ) p = 0 β 3 c = p p ULS ( ) c = L c= c= BKT 0.4 ULS crosspoint P β 7.1
79 7.3 [ ] ULS [56] c = CFT BKT massless 3.5 β= site 9,1 site 1,15 site 15,18 Central Charge P 7. β=-0.64 ( 7.1 ) c
80 β= site 9,1 site 1,15 site 15,18 Central Charge P 7.3 β= 3 ( 7.1 ) c 3.5 β= site 9,1 site 1,15 site 15,18 Central Charge P 7.4 β=-0.70 ( 7.1 ) c
81 69 8 Unknown I 1 H p,β Haldane BKT massless Unknown I Unkown I Φ (Stotal z = l) L Unknown I Stotal z = l (0 l L 1) 0 (Stotal z = l) L= (Stotal z = 0) 0(Stotal z = l) (l) 0 (Stotal z = 0) 0 (Stotal z = l) (8.1) ( ) ( (l) ) Uimin-Lai-Sutherland(ULS) (ULS crosspoint) ( triplet ) 1 ( singlet ) Unknown I 0 l L 1 0 (Stotal z = l) 8.1 Unknown I 8.1 Unknown I (OBC) 3 3 ( triplet ) 1 ( singlet ) 0 (Stotal z = l) (0 l L 1) [ 0 (Stotal z = l) (0 l L 1) < 0] l
82 70 8 Unknown I Unknown I Haldane Haldane (PBC) 1 S 1 (OBC) 1 singlet 3 triplet S Unknown I Haldane 0 (Stotal z = 0) 0(Stotal z = 1) Unknown I 0 (Stotal z = 0) 0(Stotal z = 1) Unknown I Haldan = 0 (S total z z =0) - 0 (S total = 1) = 0 (S total z z =0) - 0 (S total = ) P P β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π 8.1 L=1 (1) 8. L=1 () = 0 (S z z total =0) - 0 (S total = 3) = 0 (S z z total =0) - 0 (S total = 4) P P β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π 8.3 L=1 (3) 8.4 L=1 (4)
83 71 = 0 (S total z z =0) - 0 (S total = 5) = 0 (S total z z =0) - 0 (S total = 6) P P β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π 8.5 L=1 (5) 8.6 L=1 (6) = 0 (S total z z =0) - 0 (S total = 7) = 0 (S total z z =0) - 0 (S total = 8) P P β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π 8.7 L=1 (7) 8.8 L=1 (8) = 0 (S z z total =0) - 0 (S total = 9) = 0 (S z z total =0) - 0 (S total =10) P P β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π 8.9 L=1 (9) 8.10 L=1 (10)
84 7 8 Unknown I = 0 (S total z z =0) - 0 (S total =11) P β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π 8.11 L=1 (11) = 0 (S total z z =0) - 0 (S total = 1) = 0 (S total z z =0) - 0 (S total = ) P P β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π 8.1 L=15 (1) 8.13 L=15 () = 0 (S z z total =0) - 0 (S total = 3) = 0 (S z z total =0) - 0 (S total = 4) P P β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π 8.14 L=15 (3) 8.15 L=15 (4)
85 73 = 0 (S total z z =0) - 0 (S total = 5) = 0 (S total z z =0) - 0 (S total = 6) P P β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π 8.16 L=15 (5) 8.17 L=15 (6) = 0 (S total z z =0) - 0 (S total = 7) = 0 (S total z z =0) - 0 (S total = 8) P P β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π 8.18 L=15 (7) 8.19 L=15 (8) = 0 (S z z total =0) - 0 (S total = 9) = 0 (S z z total =0) - 0 (S total =10) P P β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π 8.0 L=15 (9) 8.1 L=15 (10)
86 74 8 Unknown I = 0 (S total z z =0) - 0 (S total =11) = 0 (S total z z =0) - 0 (S total =1) P P β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π 8. L=15 (11) 8.3 L=15 (1) = 0 (S total z z =0) - 0 (S total =13) = 0 (S total z z =0) - 0 (S total =14) P P β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π β ULS cross point L= 6 1 1/ 4π L= 9 3 3/ 7π L= 1 3 3/ 10π 8.4 L=15 (13) 8.5 L=15 (14)
87 S=1 Bilinear-Biquadratic ( S=1 BLBQ ) 1 H p,β. 1 H p,β Haldane ( ) a b Berezinskii-Kosterlitz-Thouless(BKT) massless c Unknown I 1. 1 H p,β S=1 Bilinear-Biquadratic ( S=1 BLBQ ) ( 4.1) S=1 BLBQ (ULS ) Haldane ( ). a 1 H p,β β S=1 BLBQ ( 3 < β 0) ((1 + β) 0 p (1+β) 1 4 ) b 1 H p,β Uimin-Lai-Sutherland(ULS) (p = 0 β = 3 ) ULS ( ) c = (CFT) BKT massless BKT c p = 0 β = 1 ( 9.1 ) Haldane ( ) BKT massless Unknown I L (L: ) [50]
88 Unknown I 1 H p,β p > 0 1 β < 0 Haldane ( ) ( 9. ) Haldane ( )
89
90 Haldane
91 79 S=1 Bilinear-Biquadratic KOBPACK/1 Version1.0 KOBPACK
92
93 81 [1] 1 ( ) [] F.D.M.Haldane:Phys.Lett. 93A (1983),p464 [3] K.Katsumata,H.Hori,T.Takeuchi,M.Date,A.Yamagishi and J.P.Remard:Phys.Rev.Lett. 65 (1990),p86-88 [4] J.P.Renard,M.Verdaguer,L.P.Renault,W.A.C.rkelens,J.Rossat-Mignod and W.G.Stirling:urophys.Lett. 3 (1987),p945 [5] V.Gadet,M,Verdauguer,V.Briois,A.Gleizes,J.P.Renard,P,Beaucillain,C.Chappert,T.Goto,K.Le Dang and P.Veillet:Phys.Rev.B 44 (1991),p705 [6] J.Darriet and L.P.Renault:Solid State Comm. 86 (1993),p409 [7] T.Yokoo,T.Sakaguchi,K.Kakurai and J.Akimitsu:J.Phys.Soc.Jpn 64 (1996),p3651 [8] I.Affleck,T.Keneedy,.H.Liev and H.Tasaki:Phys.Rev.Lett. 59 (1987),p799 [9] I.Affleck,T.Keneedy,.H.Liev and H.Tasaki:Commun,Math.Phys. 115 (1988),p477 [10] ( ) [11] S.Sachdev:Quantum Phase Transition(Cambridge Univ.Press,1999) [1] 1 ( ) [13] N.D.Mermin and H.Wargner:Phys.Rev.Lett. 17 (1966) p1133 [14] P.C.Hohengerg:Phys.Rev. 158 (1967) p383 [15] N.D.Mermin:J.Math.Phys. 8 (1967) p1061 [16] T.Momoi:J.Stat.Phys. 85 (1996) p193 [17] V.L.Berezinskii:Sov.Phys.JTP 3 (1970) p11;ibid 34 (1971) p493 [18] J.M.Kosterlitz and D.J.Thouless:J.Phys.C 6 (1973) p1181 [19].H.Lieb, T.D.Schultz and D.Mattis : Ann. Phys. 16 (1961) p407 [0] B.M.McCoy:Phys.Rev. 173 (1968) p531 [1] L.Hulthen:Arkiv.Mat.Astron.Fyzik 6A (1938) No.11 [] C.N.Yang and C.P.Yang:Phys.Rev. 150 (1966) 31;ibid 37 [3] M.Takahashi:Thermodynamics of One-Dimensional Solvable Models(Cambridge Univ. Press, 1999) [4] I.Affleck,D.Gepner,H.J.Schultz and T.Ziman:J.Phys.A:Math.Gen. (1989) p511 [5] R.R.Singh,M..Fisher and R.Shankar:Phys.Rev.B 39 (1989) p56 [6] A.Lunther and I.Peschel:Phys. Rev.B 1 (1975) p3908 [7] N.M.Bogoliubov,A.G.Izergin and V..Korepin:Nucl.Phys.B 75[FS17] (1986) p687 [8] T.Murashima and K.Nomura:Phys.Rev.B 73 (006),p [9] S.K.Yip:Phys.Rev.Lett. 90 (003) p 5040 [30] J.J.Garcia-Ripoll,M.A.Martin-Delgado and J.I.Cirac:Phys.Rev.Lett.93 (004) p [31] P.Millet,F.Mila,F.C.Zhang,M.Mambrini,A.B.VanOosten,V.A.Pashchenko,A.Sulpice and A.Stepanov :Phys.Rev.Lett. 83 (1999) p4176
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(iii) 0 V, x V, x + 0 = x. 0. (iv) x V, y V, x + y = 0., y x, y = x. (v) 1x = x. (vii) (α + β)x = αx + βx. (viii) (αβ)x = α(βx)., V, C.,,., (1)
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SO(3) 48 6 SO(3) t 6.1 u, v u = u 1 1 + u 2 2 + u 3 3 = u 1 e 1 + u 2 e 2 + u 3 e 3, v = v 1 1 + v 2 2 + v 3 3 = v 1 e 1 + v 2 e 2 + v 3 e 3 (6.1) i (e i ) e i e j = i j = δ ij (6.2) ( u, v ) = u v = ij
n (1.6) i j=1 1 n a ij x j = b i (1.7) (1.7) (1.4) (1.5) (1.4) (1.7) u, v, w ε x, ε y, ε x, γ yz, γ zx, γ xy (1.8) ε x = u x ε y = v y ε z = w z γ yz
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II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2
![II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2](/thumbs/101/149159646.jpg "II No.01 [n/2] [1]H n (x) H n (x) = ( 1) r n! r!(n 2r)! (2x)n 2r. r=0 [2]H n (x) n,, H n ( x) = ( 1) n H n (x). [3] H n (x) = ( 1) n dn x2 e dx n e x2")
II No.1 [n/] [1]H n x) H n x) = 1) r n! r!n r)! x)n r r= []H n x) n,, H n x) = 1) n H n x) [3] H n x) = 1) n dn x e dx n e x [4] H n+1 x) = xh n x) nh n 1 x) ) d dx x H n x) = H n+1 x) d dx H nx) = nh
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微分積分 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. ttp://www.morikita.co.jp/books/mid/00571 このサンプルページの内容は, 初版 1 刷発行時のものです. i ii 014 10 iii [note] 1 3 iv 4 5 3 6 4 x 0 sin x x 1 5 6 z = f(x, y) 1 y = f(x)
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医系の統計入門第 2 版 サンプルページ この本の定価 判型などは, 以下の URL からご覧いただけます. http://www.morikita.co.jp/books/mid/009192 このサンプルページの内容は, 第 2 版 1 刷発行時のものです. i 2 t 1. 2. 3 2 3. 6 4. 7 5. n 2 ν 6. 2 7. 2003 ii 2 2013 10 iii 1987
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(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x
![(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x](/thumbs/104/162595168.jpg "(Compton Scattering) Beaming 1 exp [i (k x ωt)] k λ k = 2π/λ ω = 2πν k = ω/c k x ωt ( ω ) k α c, k k x ωt η αβ k α x β diag( + ++) x β = (ct, x) O O x")
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2005 9/8-11 2 2.2 ( 2-5) γ ( ) γ cos θ 2πr πρhr 2 g h = 2γ cos θ ρgr (2.1) γ = ρgrh (2.2) 2 cos θ θ cos θ = 1 (2.2) γ = 1 ρgrh (2.) 2 2. p p ρgh p ( ) p p = p ρgh (2.) h p p = 2γ r 1 1 (Berry,1975) 2-6
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No δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i x j δx j (5) δs 2
No.2 1 2 2 δs δs = r + δr r = δr (3) δs δs = r r = δr + u(r + δr, t) u(r, t) (4) δr = (δx, δy, δz) u i (r + δr, t) u i (r, t) = u i δx j (5) δs 2 = δx i δx i + 2 u i δx i δx j = δs 2 + 2s ij δx i δx j
2.2 h h l L h L = l cot h (1) (1) L l L l l = L tan h (2) (2) L l 2 l 3 h 2.3 a h a h (a, h)
1 16 10 5 1 2 2.1 a a a 1 1 1 2.2 h h l L h L = l cot h (1) (1) L l L l l = L tan h (2) (2) L l 2 l 3 h 2.3 a h a h (a, h) 4 2 3 4 2 5 2.4 x y (x,y) l a x = l cot h cos a, (3) y = l cot h sin a (4) h a
SO(3) 7 = = 1 ( r ) + 1 r r r r ( l ) (5.17) l = 1 ( sin θ ) + sin θ θ θ ϕ (5.18) χ(r)ψ(θ, ϕ) l ψ = αψ (5.19) l 1 = i(sin ϕ θ l = i( cos ϕ θ l 3 = i ϕ
SO(3) 71 5.7 5.7.1 1 ħ L k l k l k = iϵ kij x i j (5.117) l k SO(3) l z l ± = l 1 ± il = i(y z z y ) ± (z x x z ) = ( x iy) z ± z( x ± i y ) = X ± z ± z (5.118) l z = i(x y y x ) = 1 [(x + iy)( x i y )
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v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) 3 R ij R ik = δ jk (4) i=1 δ ij Kronecker δ ij = { 1 (i = j) 0 (i
1. 1 1.1 1.1.1 1.1.1.1 v v = v 1 v 2 v 3 (1) R = (R ij ) (2) R (R 1 ) ij = R ji (3) R ij R ik = δ jk (4) δ ij Kronecker δ ij = { 1 (i = j) 0 (i j) (5) 1 1.1. v1.1 2011/04/10 1. 1 2 v i = R ij v j (6) [
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1. k λ ν ω T v p v g k = π λ ω = πν = π T v p = λν = ω k v g = dω dk 1) ) 3) 4). p = hk = h λ 5) E = hν = hω 6) h = h π 7) h =6.6618 1 34 J sec) hc=197.3 MeV fm = 197.3 kev pm= 197.3 ev nm = 1.97 1 3 ev
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V(x) m e V 0 cos x π x π V(x) = x < π, x > π V 0 (i) x = 0 (V(x) V 0 (1 x 2 /2)) n n d 2 f dξ 2ξ d f 2 dξ + 2n f = 0 H n (ξ) (ii) H
199 1 1 199 1 1. Vx) m e V cos x π x π Vx) = x < π, x > π V i) x = Vx) V 1 x /)) n n d f dξ ξ d f dξ + n f = H n ξ) ii) H n ξ) = 1) n expξ ) dn dξ n exp ξ )) H n ξ)h m ξ) exp ξ )dξ = π n n!δ n,m x = Vx)
.2 ρ dv dt = ρk grad p + 3 η grad (divv) + η 2 v.3 divh = 0, rote + c H t = 0 dive = ρ, H = 0, E = ρ, roth c E t = c ρv E + H c t = 0 H c E t = c ρv T
NHK 204 2 0 203 2 24 ( ) 7 00 7 50 203 2 25 ( ) 7 00 7 50 203 2 26 ( ) 7 00 7 50 203 2 27 ( ) 7 00 7 50 I. ( ν R n 2 ) m 2 n m, R = e 2 8πε 0 hca B =.09737 0 7 m ( ν = ) λ a B = 4πε 0ħ 2 m e e 2 = 5.2977
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A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B
9 7 A = A x x + A y y + A, B = B x x + B y y + B, C = C x x + C y y + C..6 x y A B C = A x x + A y y + A B x B y B C x C y C { B = A x x + A y y + A y B B x x B } B C y C y + x B y C x C C x C y B = A
1 1.1 H = µc i c i + c i t ijc j + 1 c i c j V ijklc k c l (1) V ijkl = V jikl = V ijlk = V jilk () t ij = t ji, V ijkl = V lkji (3) (1) V 0 H mf = µc
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4. ϵ(ν, T ) = c 4 u(ν, T ) ϵ(ν, T ) T ν π4 Planck dx = 0 e x 1 15 U(T ) x 3 U(T ) = σt 4 Stefan-Boltzmann σ 2π5 k 4 15c 2 h 3 = W m 2 K 4 5.
A 1. Boltzmann Planck u(ν, T )dν = 8πh ν 3 c 3 kt 1 dν h 6.63 10 34 J s Planck k 1.38 10 23 J K 1 Boltzmann u(ν, T ) T ν e hν c = 3 10 8 m s 1 2. Planck λ = c/ν Rayleigh-Jeans u(ν, T )dν = 8πν2 kt dν c
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Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 = ( p µ γ µ + m)(p ν γ ν + m) (5.1) γ = p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 = 1 2 p µp ν {γ µ, γ ν } + m
Dirac 38 5 Dirac 4 4 γ µ p µ p µ + m 2 p µ γ µ + mp ν γ ν + m 5.1 γ p µ p ν γ µ γ ν p µ γ µ m + mp ν γ ν + m 2 1 2 p µp ν {γ µ, γ ν } + m 2 5.2 p m p p µ γ µ {, } 10 γ {γ µ, γ ν } 2η µν 5.3 p µ γ µ + mp
[ ] (Ising model) 2 i S i S i = 1 (up spin : ) = 1 (down spin : ) (4.38) s z = ±1 4 H 0 = J zn/2 i,j S i S j (4.39) i, j z 5 2 z = 4 z = 6 3
![[ ] (Ising model) 2 i S i S i = 1 (up spin : ) = 1 (down spin : ) (4.38) s z = ±1 4 H 0 = J zn/2 i,j S i S j (4.39) i, j z 5 2 z = 4 z = 6 3](/thumbs/103/161261933.jpg "[ ] (Ising model) 2 i S i S i = 1 (up spin : ) = 1 (down spin : ) (4.38) s z = ±1 4 H 0 = J zn/2 i,j S i S j (4.39) i, j z 5 2 z = 4 z = 6 3")
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7 7. ( ) SU() SU() 9 ( MeV) p 98.8 π + π 0 n 99.57 9.57 97.4 497.70 δm m 0.4%.% 0.% 0.8% π 9.57 4.96 Σ + Σ 0 Σ 89.6 9.46 K + K 0 49.67 (7.) p p = αp + βn, n n = γp + δn (7.a) [ ] p ψ ψ = Uψ, U = n [ α
1 1.1 ( ). z = a + bi, a, b R 0 a, b 0 a 2 + b 2 0 z = a + bi = ( ) a 2 + b 2 a a 2 + b + b 2 a 2 + b i 2 r = a 2 + b 2 θ cos θ = a a 2 + b 2, sin θ =
1 1.1 ( ). z = + bi,, b R 0, b 0 2 + b 2 0 z = + bi = ( ) 2 + b 2 2 + b + b 2 2 + b i 2 r = 2 + b 2 θ cos θ = 2 + b 2, sin θ = b 2 + b 2 2π z = r(cos θ + i sin θ) 1.2 (, ). 1. < 2. > 3. ±,, 1.3 ( ). A
i
009 I 1 8 5 i 0 1 0.1..................................... 1 0.................................................. 1 0.3................................. 0.4........................................... 3
() x + y + y + x dy dx = 0 () dy + xy = x dx y + x y ( 5) ( s55906) 0.7. (). 5 (). ( 6) ( s6590) 0.8 m n. 0.9 n n A. ( 6) ( s6590) f A (λ) = det(a λi)
0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
1 1 sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω 1 ω α V T m T m 1 100Hz m 2 36km 500Hz. 36km 1
sin cos P (primary) S (secondly) 2 P S A sin(ω2πt + α) A ω ω α 3 3 2 2V 3 33+.6T m T 5 34m Hz. 34 3.4m 2 36km 5Hz. 36km m 34 m 5 34 + m 5 33 5 =.66m 34m 34 x =.66 55Hz, 35 5 =.7 485.7Hz 2 V 5Hz.5V.5V V
2009 2 26 1 3 1.1.................................................. 3 1.2..................................................... 3 1.3...................................................... 3 1.4.....................................................
29
9 .,,, 3 () C k k C k C + C + C + + C 8 + C 9 + C k C + C + C + C 3 + C 4 + C 5 + + 45 + + + 5 + + 9 + 4 + 4 + 5 4 C k k k ( + ) 4 C k k ( k) 3 n( ) n n n ( ) n ( ) n 3 ( ) 3 3 3 n 4 ( ) 4 4 4 ( ) n n
128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds = 0 (3.4) S 1, S 2 { B( r) n( r)}ds
127 3 II 3.1 3.1.1 Φ(t) ϕ em = dφ dt (3.1) B( r) Φ = { B( r) n( r)}ds (3.2) S S n( r) Φ 128 3 II S 1, S 2 Φ 1, Φ 2 Φ 1 = { B( r) n( r)}ds S 1 Φ 2 = { B( r) n( r)}ds (3.3) S 2 S S 1 +S 2 { B( r) n( r)}ds
2 1 κ c(t) = (x(t), y(t)) ( ) det(c (t), c x (t)) = det (t) x (t) y (t) y = x (t)y (t) x (t)y (t), (t) c (t) = (x (t)) 2 + (y (t)) 2. c (t) =
1 1 1.1 I R 1.1.1 c : I R 2 (i) c C (ii) t I c (t) (0, 0) c (t) c(i) c c(t) 1.1.2 (1) (2) (3) (1) r > 0 c : R R 2 : t (r cos t, r sin t) (2) C f : I R c : I R 2 : t (t, f(t)) (3) y = x c : R R 2 : t (t,