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2 (Onsager )

3 1 2 2 Onsager A 27 A A A

4 1 Onsager [1] Onsager Kaufan [2] Kasteleyn [3][4] Kaufan Kastening Kastening SO(2,C) NbSe 3 (1.1)[5] [6] 0 (1.2) s +1 = s 1 1.1: a. b. c. (Tanda et al.[5]) 2

5 1.2: Kastening a = J/kT (a) (a ) Z(a ) sinh N 2 (2a ) = Z(a) sinh N 2 (2a) a C e 2a C =

6 2Onsager Kastening Onsager Kastening l (2.1) E = E a + E b (2.1) E a = J a E b = J b l λ=1 l λ=1 s λ µ s λ+1,µ (2.2) s λ µ s λ,µ+1 (2.3) β 1 = kt j a j b 1 l a b a b l a a b b a a b b a b a b b b 2.1: a b l s λ µ ±1 a = βj a b = βj b 4

7 Z(a,b) = s 11 s l exp( βe) (2.4) 2 n 2 n T Z(a,b) = TrT T π T π = exp(as µ s µ + bs µ s µ+1 ) (2.5) ν = 1, l π λ = {s λ1,,s λ } T T = V b V a V a V b π V a π = π V b π = exp(as µ s µ) (2.6) δ sµ s µ exp(bs µs µ+1 ) (2.7) σ x σ y σ z X µ = } 1 {{ 1 } σ x } 1 {{ 1 } µ 1 µ (2.8) 2 2 Y µ Z µ σ y σ z sinh(2ā)sinh(2a) = 1 ā > 0 V a = [2sinh(2a)] l/2 V a (2.9) V b = V a = n exp(āx µ ) (2.10) exp(bz µ Z µ+1 ) (2.11) Z +1 = Z 1 T = [2sinh(2a)] l/2 V a V b (2.12) Z(a,b) = [2sinh(2a)] l/2 TrV (2.13) V V = V a/2 V b V a/2 (2.14) 5

8 n V a/2 = exp(āx µ /2) (2.15) V 2 a/2 = V a V 2 Λ k Z(a,b) = [2sinh(2a)] l/2 2 Λ k (2.16) V {Γ λ,γ µ } = 2δ λ µ (2.17) Γ 2µ 1 = X 1 X µ 1 Z µ (2.18) Γ 2µ = X 1 X µ 1 Y µ (2.19) 2 ( µ = 1,, ) U X = X 1 X n = i n Γ 1 Γ 2 Γ 2 n, U 2 X = 1 (2.20) U X Γ λ {Γ λ,u X } = 0 (2.11) (2.15) X µ = i 2 [Γ 2µ,Γ 2µ 1 ], µ = 1,, (2.21) Z µ Z +1 = i 2 [Γ 2µ+1,Γ 2µ ], µ = 1,, 1 (2.22) Z Z 1 = i 2 U X[Γ 1,Γ 2 ] (2.23) Kaufan [2] Huang [7] Γ µ U X Γ α Γ β (2.11) (2.15) (2.14) V V P ± = 1 2 (1 ± U X) (2.24) J αβ = i 4 [Γ α,γ β ] (2.25) J ± αβ =P± J αβ (2.26) J αβ = J + αβ + J αβ (2.27) U X J ± αβ = ±J± αβ (2.28) 6

9 J ± αβ = J± βα J+ αβ J αβ (2 1) [J ± αβ,j± γδ ] = i(δ αγj ± βδ + δ βδ J ± αγ δ αδ J ± βγ δ βγj ± αδ ) (2.29) (2.22)(2.23) (2.25)(2.28) X µ = 2(J + 2µ,2µ 1 + J 2µ,2µ 1 ), µ = 1,, (2.30) Z µ Z +1 = (J + 2µ+1,2µ + J 2µ+1,2µ ), µ = 1,, 1 (2.31) Z Z 1 = 2U X (J + 1,2 + J 1,2 ) = 2(J+ 1,2 + J 1,2 ) (2.32) (2.15) V a/2 J ± αβ V a/2 = n exp[ā(j + 2µ,2µ 1 + J 2µ,2µ 1 )] = V+ a/2 V a/2 (2.33) V ± n a/2 = exp(āj ± 2µ,2µ 1 ) (2.34) (2.11) V b V b = exp[ 2b(J + 1 1,2 J 1,2 )] exp[2b(j + 2µ+1,2µ + J 2µ+1,2µ )] = V+ b V b (2.35) V ± 1 b = exp( 2bJ± 1,2 ) exp(2bj ± 2µ+1,2µ ) (2.36) (2.14) V V ± = V ± a/2 V± b V± a/2, [V+,V ] = 0 (2.37) V = V + V M M J αβ (J αβ ) i j = i(δ αi δ β j δ βi δ α j ) (2.38) 1 M j αβ = J βα M(M 1)/2 M = 2 (2.29) c αβ S = exp(ic αβ J αβ ) S S T = S 1 dets = 1 M M SO(M,C) 7

10 (2.37) V ± (2.34) V ± a/2 (2.36) V± b SO(2,C) V ± = V ± a/2 V b ± V ± a/2 (2.39) V ± a/2 = n exp(āj 2µ,2µ 1 ± ) (2.40) V ± b 1 = exp( 2bJ± 1,2 ) exp(2bj ± 2µ+1,2µ ) (2.41) ā b V ± a/2 V b ± V ± V ± V ± S ± V S ± = S ±V ± S± 1 ) V ± S = exp ( γ { 2µ 1 2µ 2} J 2µ,2µ 1 γ k (2.42) coshγ k = cosh2ācosh2b cos πk sinh2āsinh2b (2.43) b = 0 γ k = 2ā b γ k k = 1,,2 1 γ k > 0 k = 0 γ 0 = 2(ā b) (2.44) γ k γ 0 ā > b ā < b γ k V S ± 2 2 V ± S (2.42) S ± SO(2n,C) S ± = exp(ic ± αβ J αβ ) c ± αβ (2 2 ) S = S + S S ± = exp(ic ± αβ J αβ ) (2.45) V S = SVS 1 = S + V + S 1 + S V S 1 V + S V S (2.46) 8

11 ā b c ± αβ V± S J αβ ā b c ± αβ V S ± J αβ Bakaer- Capbell-Hausdorff ( exp(a) exp(b) = exp A + B [A,B] + 1 { [A,[A,B] ] [ ] } + [A,B],B + ) 12 (2.47) V ± S ] V ± S = exp [ γ { 2µ 1 2µ 2} J 2µ,2µ 1 V S = exp = exp [ (γ 2µ 1 J 2µ,2µ 1 + γ 2µ 2 J 2µ,2µ 1 ] (2.48) [ 1 ( ) ] 4 γ2µ 1 (1 + U X )X µ + γ 2µ 2 (1 U X )X µ (2.49) V ± S V Y = R Y V S R 1 Y R Y R ±1 Y = 2 n/2 (1 ± iy µ ) (2.50) R Y X µ R 1 = Z µ V S Y V Y = R Y V S R 1 Y [ 1 ( ) ] = exp 4 γ2µ 1 (1 + U Z )Z µ + γ 2µ 2 (1 U Z )Z µ (2.51) U Z = Z 1 Z µ V Y U Z +1 1 ( ) Z µ 1 U Z +1( 1) (i) ( ) Z µ 1 (1 + U Z )/2 1(0) (ii) ( ) Z µ 1 (1 U Z )/2 1(0) V ( ) 1 exp (±)γ 2µ 1 (2.52) ( 1 exp 2 (±)γ 2µ 2 ) (2.53) 9

12 (±) (2.52) ( ) (2.53) (e) (o) (2.16) Z ( ) Z(a,b) = [2sinh(2a)] { l/2 l exp e 2 (±)γ 2µ 1 ( )} l + exp o 2 (±)γ 2µ 2 (2.54) [ )] [ )] 2cosh( 2 γ 2k 1 + 2sinh( 2 γ 2k 1 = 1 2 [2sinh(2a)]l/2 [ k=1 + k=1 k=1 [ )] 2cosh( 2 γ 2k 2 k=1 [ )] 2sinh( ] 2 γ 2k 2 (2.55) 10

13 3 Kastening 3.1 l ( ( ) 1 βe = a σ a s lµ s 1µ + s λ µ s λ+1,µ )+b σ b s λ s λ1 + s λ µ s λ,µ+1 l 1 λ=1 l λ=1 (3.1) a,b > 0 s λ µ ±1 (σ a,σ b ) = (1,1) (σ a,σ b ) = (1, 1) ( 1,1) ( 1, 1) (pa) (ap) (aa) Z (l,) αβ (a,b) = exp( β E) (3.2) s λ µ =±1 αβ pppaap aa X µ (Y µ Z n ) X µ = } 1 {{ 1 } σ x } 1 {{ 1 } µ 1 µ (3.3) σ x Z (l,) αβ (a,b) = [2sinh(2a)]l/2 Tr(QV l ) (3.4) 11

14 a Q = 1 Q = U X X 1 X 2 X n V = V a/2 V b Va/2 (3.5) V a/2 = exp(āx µ /2) (3.6) 1 V b = exp(σ b bz µ Z µ+1 ) exp(bz µ Z λ+1 ) (3.7) ā sinh2asinh2ā = 1 b σ b = 1 σ b = 1 J ± αβ = P± J αβ (3.8) P ± = 1 2 (1 ± U X) (3.9) J αβ = i 4 [Γ α,γ β ] (3.10) Γ 2µ 1 = X 1 X µ 1 Z µ (3.11) Γ 2µ = X 1 X µ 1 Y µ (3.12) Z (l,) αβ (a,b) = [2sinh(2a)]l/2 Tr(Q a V l β ) (3.13) Q { p a} =P + ± P (3.14) V β = V + β V β = V b V+ β (3.15) V ± β = V a/2 ± V ± bβ V± a/2 (3.16) V ± a/2 = exp(āj ± 2µ,2µ 1 ) (3.17) V ± bp = exp( 2bJ± 1,2 ) exp(2bj ± 2µ+1,2µ ) (3.18) V ± ba = exp(±2bj± 1,2 ) exp(2bj ± 2µ+1,2µ ) (3.19) γ k coshγ k = cosh2ācosh2b cos πk sinh2āsinh2b (3.20) 1 k γ k > 0 γ 0 = 2(ā b) b () () γ k a 12

15 ( ) Σ (l,) l ee (a,b) = exp e 2 (±)γ 2µ 2 (3.21) ( ) Σ eo (l,) l (a,b) = exp e 2 (±)γ 2µ 1 (3.22) ( ) Σ oe (l,) l (a,b) = exp o 2 (±)γ 2µ 2 (3.23) ( ) Σ (l,) l oo (a,b) = exp (±)γ 2µ 1 (3.24) 2 o Σ xy x ( ) y γ k ( ) C o (l,) (a,b) = [2sinh(2a)] l/2 [ ( )] l 2cosh k=1 2 γ 2k 1 (3.25) S o (l,) (a,b) = [2sinh(2a)] l/2 [ ( )] l 2sinh k=1 2 γ 2k 1 (3.26) C e (l,) (a,b) = [2sinh(2a)] l/2 [ ( )] l 2cosh k=1 2 γ 2k 2 (3.27) S e (l,) (a,b) = [2sinh(2a)] l/2 [ ( )] l 2sinh 2 γ 2k 2 (3.28) S (2l,) o S (2l,) e k=1 (a,b) = S o (l,) (a,b)c o (l,) (a, b) (3.29) (a,b) = S e (l,) (a,b)c e (l,) (a, b) (3.30) 13

16 (l,) (a,b) Z pp = [2sinh(2a)] l/2 (Σ eo + Σ oe ) = 1 2 (C o + S o +C e S e ) (3.31) Z pa = [2sinh(2a)] l/2 (Σ ee + Σ oo ) = 1 2 (C o S o +C e + S e ) (3.32) Z ap = [2sinh(2a)] l/2 (Σ eo + Σ oe ) = 1 2 (C o + S o C e + S e ) (3.33) Z aa = [2sinh(2a)] l/2 (Σ ee + Σ oo ) = 1 2 ( C o + S o +C e + S e ) (3.34) 3.2 logz/(βl) β f pp = 1 l logz pp = 1 l log 1 2 (C o + S o +C e S e ) = 1 2 log(2sinh2a) [ + 1 l log 1 2 i + j = 1 2 log(2sinh2a) [ + 1 l log 1 2 i + j (e α i + e α i ) + (e α j + e α j ) i (e α i e α i ) j (e α j e α j ) e α i (1 + e 2α i ) +e α i (1 e 2α i ) i e α j (1 + e 2α j ) e α j (1 e 2α j ) j 14 ] ]

17 α k = lγ k /2 i = 1,3,,2 1 j = 0,2,,2 2 (3.36) (3.41)(3.42)(3.43) l l log(a + x) = loga(1 + x a ) = loga + x a 1 2 ( x a )2 (a x) (3.35) β f pp = 1 2 log(2sinh2a) + li 1 2 li γ 2k 1 + li k=1 1 l e lγ 2k 2 (3.36) k=1 γ(ν) =γ 2k 1 (3.37) ν = π (2k 1) (3.38) k=1γ 2k 1 = 2π β f pp = 1 2 log(2sinh2a) + 1 2π π 0 2π 0 dνγ(ν) (3.39) dνγ(ν) + 1 π dνe lγ(ν) (3.40) lπ 0 γ (2.43) γ(2π ν) = γ(ν) β f pa = 1 log(2sinh2a) + li 2 = 1 2 log(2sinh2a) + 1 2π 1 2 π β f ap = 1 log(2sinh2a) + li 2 = 1 2 log(2sinh2a) + 1 2π π β f aa = 1 log(2sinh2a) + li 2 = 1 2 log(2sinh2a) + 1 2π π 0 γ 2k 2 + li k=1 dνγ(ν) + 1 π lπ 0 γ 2k 1 li k=1 dνγ(ν) 1 π lπ 0 γ 2k 2 li k=1 dνγ(ν) 1 lπ π 0 1 l e lγ 2k 1 k=1 dνe lγ(ν) (3.41) 1 l e lγ 2k 2 k=1 dνe lγ(ν) (3.42) 1 l e lγ 2k 1 k=1 dνe lγ(ν) (3.43) (3.39) (3.37) (3.38) (2k 1) (2k 2) (3.36) (3.41) 15

18 : β f p a : l : β f p a () (3.40) (3.42) (3.43) β f p a = β( f pp f ap ) = 2 π dνe lγ(ν) (3.44) lπ 0 (l) l (b = a) γ(ν) (3.20) coshγ k = cosh2acoth2a cosν (3.45) u = (β f ) (3.46) β 16

19 u = β (β f ) = 4 π jcosh2a( coth 2 2a 2 ) π 0 dν e lγ(ν) sinhγ (3.47) 4.1 cosh2a c = 2, sinh2a c = 1 (3.48) u = 0 (3.49) C = u T (3.50) [ C = 8 π ka2 cosh 2 2a ( coth 2 2a 2 ) 2 π dν e lγ(ν) 0 sinh 2 (l + cothγ) γ ] 1 π sinh 3 2a (sinh2 2acosh 2 2a + 2) dν e lγ(ν) (3.51) 0 sinhγ 17

20 4 [9] 4.1 [8] (b = a) Z { Z(a) = exp σ N =±1 σ 1 =±1 } a σ i σ j i j a = J/kT i j i j N a Z(a) σ ±1 Z(a) = σ N =±1 (4.1) exp(aσ) = cosha + σ sinha (4.2) σ 1 =±1 i j = (cosha) 2N σ N =±1 (cosha + σ i σ j sinha) σ 1 =±1 i j (1 + σ i σ j tanha) (4.3) Z(a)/cosh 2N a tanha σ = 0, σ=±1 1 = 2 (4.4) σ=±1 (1 + σ i σ j tanha) i j σ i σ j tanha 18

21 4.1: (Kogut[8]) σ i 4.1 Z(a)(cosha) 2N 2 2N = 1 + N(tanha) 4 + 2N(tanha) N(N 5)(tanha)8 + (4.5) N Z(a) T = T = Z(a)e 2Na = 1 + Ne 8a + 2Ne 12a N(N 5)e 16a + (4.6) ( ) a tanha = exp( 2a ) (4.7) (4.5) (4.6) Z(a ) (e 2a ) N = Z(a) 2 N (cosh 2 a) N (4.8) 4.2: 19

22 (4.7) a a ( ) (4.7) (4.8) sinh(2a)sinh(2a ) = 1 (4.9) Z(a ) sinh N 2 (2a ) = Z(a) sinh N 2 (2a ) (4.10) a = a (4.9) sinh 2 (2a C ) = 1 (4.11) e 2a C = (4.12) [9] 4.2 l ( ) N = l 2N σ +1 = σ 1 (4.13) E = a σ i σ j + a σ p σ q (4.14) i j pq pq + 1 = 1 l i j 20

23 4.3: (Kogut[8]) Z { } Z(a) = exp a σ i σ j a σ p σ q σ N =±1 σ 1 =±1 i j pq = (cosha + σ i σ j sinha)(cosha σ p σ q sinha) σ N =±1 σ 1 =±1 i j, pq = (cosha) 2N (1 + σ i σ j tanha)(1 σ p σ q tanha) σ N =±1 σ 1 =±1 i j, pq (4.15) + 1 (4.3) (4.5) f E = a(2n f ) + a f = 2a(N f ) (4.16) l E = 2a(N l) (4.4) 4.4: 2 21

24 4.5: l l E = 2a(N l) (4.5) E = 2a(N l) + 4a l E = 2a(N l) 4a l 1 l 2l(l 1) E = 2a(N l)+8a (4.6) l( 2) (4.7) E = 2a(N l) + 8a l(l 1)(l 2)(l 3)/6 ( 4.8) E = 2a(N l) + 8a l(l 1)l(l + 1)/3 22

25 [ Z = 2e 2a(N l) 1 + l(l 1)e 4a + { l( 2) l(l 1)(l2 l + 2) } ] e 8a + (4.17) (4.5) (4.17) 4.6: 23

26 4.7: 4.8: 24

27 5 (l) (3.44) (3.47) (3.51) (3.49) 25

28 26

29 A A.1 [8] ( ) j µ τ ( j, µ) ( j + µ, µ) ( ) S = J σ 3 ( j, µ)σ 3 ( j + µ,τ)σ 3 ( j + µ + τ, µ)σ 3 ( j + ν, τ) j (A.1) ( ) ( ) (σ 3 = ±1) G( j) G( j) j σ 3 ( j,τ) = 1 (A.2) l Z = spincon f ig e S (A.3) S = J σ 3 ( j, µ)σ 3 ( j + τ, µ) n (A.4) exp(βσ 3 σ 3 ) = coshβ + σ 3 σ 3 sinhβ = coshβ(1 + σ 3 σ 3 tanhβ) (A.5) Z = spincon f ig coshβ(1 + σ 3 ( j, µ)σ 3 ( j + τ, µ)tanhβ) j (A.6) 27

30 σ 3 = 0, σ 3 =±1 σ 3 =±1 σ 3 2 = 1 = 2 (A.7) σ 3 =±1 σ 3 ( j)σ 3 ( j + τ) l ( ) σ 3 tanhβ 1 Z 1 = 2 l cosh l β(1 + tanh l β) (A.8) Z = Z 1 = [2 l (cosh l β + sinh l β)] (A.9) A.2 σ 3 ( j,τ) = 1 (A.10) l T R C σ 3 = σ 3 (0,0; µ)σ 3 (0,1; µ) σ 3 (0,R; µ) σ 3 (T,0; µ)σ 3 (T,1; µ) σ 3 (T,R; µ) C (A.11) ( ) C σ 3 = spincon f ig σ 3 e S / Z C (A.12) S = J σ 3 ( j, µ)σ 3 ( j + τ, µ) j (A.13) exp(βσ 3 σ 3 ) = coshβ + σ 3 σ 3 sinhβ = coshβ(1 + σ 3 σ 3 tanhβ) (A.14) 28

31 C σ 3 = spincon f ig C σ 3 coshβ(1 + σ 3 ( j, µ)σ 3 ( j + τ, µ)tanhβ) j (A.15) σ 3 = 0, σ 3 =±1 σ 3 =±1 σ 3 2 = 1 = 2 (A.16) σ 3 =±1 σ 3 (n) c σ 3 tanh T R β tanh (l T ) R β ( ) (1 + tanh l β) R (2coshβ) l (tanh T R β + tanh (l T ) R β)(1 + tanh l β) R (A.17) Z = [2 l (cosh l β + sinh l β)] (A.18) σ 3 = (tanht R β + tanh (l T ) R β) C (1 + tanh l (A.19) β) R β tanhβ β T l A = T R σ 3 tanh A β = exp{(ln tanhβ) A} (A.20) C β 5 (A.20) λ 1,(λ 1) σ 3 ( j, µ)σ 3 ( j + τ, µ) = 1 e 4β ( ) Z = e nβ (1 + ne 4β + ) (A.21) 29

32 T R σ 3 = e nβ {1 + (n 4A)e 4β + }/ Z exp( 4Ae 4β ) (A.22) C n = l,a = (T 1) R A.3 (A.1) S = σ 3 (n, µ)σ 3 (n + µ,ν)σ 3 (n + µ + ν, µ)σ 3 (n + ν, ν) n,µν (A.23) ( ) ( ) Ising (σ 3 = ±1) ( ) G( j) G( j) j S σ 3 ( j,τ) = 1 (A.24) x +1 G 30

33 [1] L. Onsager, Phys. Rev. 65, 117 (1944). [2] B. Kaufan, Phys. Rev. 76, 1232 (1949). [3] P.W. Kasteleyn, Math. Phys (1963). [4] P.W. Kasteleyn, Physica (1961). [5] S. Tanda, T. Tsuneta, Y. Okajia, K. Inagaki, K. Yaaya, and N. Hatakenaka Nature 417, 397 (2002). [6] K. Kaneda and Y. Okabe, Phys. Rev. Lett. 86, 2134 (2001) [7] K. Huang, Statistical Mechanics, 2nd ed. (Willey) [8] J. B. Kogut, Rev. Mod. Phys. 51, 659 (1979) [9] H. A. Kraers and G. H. Wannier, Phys. Rev. 60, 252 (1941). 31

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O x y z O ( O ) O (O ) 3 x y z O O x v t = t = 0 ( 1 ) O t = 0 c t r = ct P (x, y, z) r 2 = x 2 + y 2 + z 2 (t, x, y, z) (ct) 2 x 2 y 2 z 2 = 0 9 O y O ( O ) O (O ) 3 y O O v t = t = 0 ( ) O t = 0 t r = t P (, y, ) r = + y + (t,, y, ) (t) y = 0 () ( )O O t (t ) y = 0 () (t) y = (t ) y = 0 (3) O O v O O v O O O y y O O v P(, y,, t) t (, y,, t )

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