[ ] (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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1 [ ] (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 S i S j (4.39) i, j z 5 2 z = 4 z = 6 3 z = 6 z = 8 zn/2 1 2 N i z nearest neighbors of i j=1 (4.40) J > 0 4 Ṡi = (i/ h)[h, S i ] 5 55

2 h H = H 0 + H ext = J zn/2 S i S j h N S i (4.41) i [ ] h = 0 2 (k B T J) (paramagnetism) (ferromagnetism) (spontaneous symmetry breaking) (order parameter) (magnetization) 6 M = i S i. (4.42) 7 1 m = M N = S i (4.43) i F = E T S (4.44) M = 0 8 (k B T J) 6 7 (4.48) 8 m (4.64) 56

3 M N or m 1 (4.45) (4.74) M = 0 ( ( ) ) M (phase transition) [Ising ] 1 ( ) 2 (Onsager) 3 Ising (4.74) Z(T, h) = n e βe n = 2 N n e βh n n = e βh({s i}) S 1 =±1 S 2 =±1 S N =±1 (4.46) n : n = 2 N (mean field approximation) S i S i δs i = S i S i H 0 = J S i S j 57

4 = J [(S i S i )(S j S j ) + S i S j + S i S j S i S j ](4.47) δs 2 1 S i = n n S i e βh n n n e βh n (4.48) H 0 H mf = J i n.n. of i S i j S j J i n.n. of i j S i S j = zjm i S i NzJm2 (4.49) Z = e βh eff(s 1 +S 2 + +S N ) e 1 2 βnzjm2 = S 1 =±1 S N =±1 e βh effs 1 e βh effs N e 1 2 βnzjm2 S 1 =±1 S N =±1 = ( ) e βh eff + e βh N eff e 1 2 βnzjm2 = [ e 1 2 βzjm2 2 cosh (βh eff ) ] N (4.50) h eff = h + zjm (4.51) [ 1 F (V, T, h) = N 2 zjm2 1 ] β ln [2 cosh(βh eff)] (4.52) 9 m = 1 N F h = tanh(βh eff) (4.55) 9 h i Z(β, {h i }) = n n e βĥ0 i h i S i n F (β, {h i }) = 1 β ln Z(β, {h i}) (4.53) h i h α i mα i 58

5 (4.49) m = S i m m = tanh [β(h + zjm)] (4.56) (self-consistency equation) (4.56) h = 0 T c zj k B (4.57) ( ) Tc m = tanh T m (4.58) x = (T c /T )m y = tanh(x) y = (T/T c )x (self-consistent) ( 4.5(a)) = 0 for T > zj/k B, m (4.59) = 0, ±m s (T ) for T < zj/k B T T c x = 0 ( ) T T c 3 m = 0 m = ±m s T c (critical temperature) h = 0 F = N β ln [2 cosh (βzjm s(t ))] znjm s(t ) 2 (4.60) ( h i = h ) m α i = Si α n = n Sα i e β(ĥ0 hi Si) i n h i S n n e β(ĥ0 i i) n = 1 Z (βh α i ) n e β(ĥ0 h i S i i) n = 1 β n (h α i ) ln Z(β, {h i}) (4.55) 1 h i = h = h α F (β, {h i }). (4.54) i 59

6 m Β, tanh Βm Βm f m (a) 1.0 (b) m 4.5: (a) (4.56) zj = 1 β = 0.5, 1, 2 (b) Ising f(t, m) (a) m s (4.60) 4.5(b) (4.56) ( (4.60) ) m = 0 m = ±m s (4.56) m s (T ) E = d(βf ) dβ = N tanh (βzjm s ) zjm s znjm2 s + ( NβzJ tanh (βzjm s ) + βznjm s ) dm s dβ = 1 2 znjm2 s (4.61) (tanh (βzjm s ) = m s ) (4.49) E = [ ] F (β, M) F 60

7 F (β, M) 1 β ln states of given M n e βh 0 n (4.62) M(= Nm) N + = (N + M)/2 N = (N M)/2 W (N, M) = N C N+ = N! N +!N! = N! N +!(N N + )! S k B = ln W = N ln N e N + ln N + e (N N +) ln N N + e = N + ln N + N (N N +) ln N N + N ( N+ = N N ln N + N + N N + N ( 1 + m = N ln 1 + m + 1 m ln N N + N ) ln 1 m 2 ) (4.63) (4.64) JS i S j M = J S i M S j M = Jm 2 (4.65) M M zn2 E T S F (β, m) = 1 2 znjm2 + N ( 1 + m β 2 E(m) = 1 2 znjm2 (4.66) ln 1 + m m 2 ln 1 m ) 2 (4.67) 4.6(b) [ ] 10 - (Bragg-Williams) 61

8 m Β, tanh Βm Βm f m (a) 1.0 (b) m 4.6: (a) (4.56) ( 4.5 ) zj = 1 β = 0.5, 1, 2 (b) ( - ) f(t, m) (a) 4.5 F 1 f = F/N m f = zjm + 1 2β ln 1 + m 1 m = 0 (4.68) βzjm = 1 2 ln 1 + m 1 m (4.69) m 1 + m 1 m = e2βzjm e βzjm (1 + m) = e βzjm (1 m) (4.58) [ ] m = tanh (βzjm) (4.70) 4.6(b) f(t, m) m = 1/2 m = 0 T = T c f (0) > 0 f (0) < 0 f(t, m) m = 0 T < T c 3 4.7(a) (b) 4.7(c) f(t, m) T = T c 3 62

(a) 1.0 1.0 0.5 0.5 m 0.0 m 0.0 0.5 0.5 (b) 1.0 0.0 0.5 1.0 1.5 2.0 T Tc (c) 1.0 0.0 0.5 1.0 1.5 2.0 T Tc 4.

9 (a) m 0.0 m (b) T Tc (c) T Tc 4.7: (a) f(t, m) (b) f(t, m) (c) f(t, m) () m = ±1 m = ± [ ] Ising Ĥ 0 = J n.n. S i S j (4.71) 63

10 (a) (b) (c) 4.8: F (T, m) (a) T > T c (b)t = T c (c) T < T c F 2 (XY ) (m x, m y ) F (T, m) T c 4.8 m = 0 T < T c 4.8(c) Bourgogne Bordeaux m = 0 m = m 0 m = 0 ( Hilbert ) m ( 0) ( Ising Heisenberg 2 64

11 XY ) 2 Ising XY 2 T < T c Berezinskii-Kosterlitz-Thouless(BKT) [ ] [ ] 2 A B A B 2a 2a (sublattice) J J < 0 A B J > 0 (4.74) J < 0 zn/2 Ĥ0 AF = J Si A Sj B (4.72) σ i = S A i, σ j = S B j, J = J (> 0) (4.73) zn/2 zn/2 Ĥ0 AF = J σ i ( σ j ) = J σ i σ j (4.74) J > 0 S B j J < 0 A B (antiferro magnetism) m A = 2 N N A + N A, m B = 2 N N B + N B. (4.75) 65

12 m A = m B m m = 0 m 0 T N = 1 k B z J (4.76) (Neel) [ ] (lattice gas model) i n i n i = 1 n i = 0 ϕ (< 0) H = ϕ n i n j (4.77) ( ) 11 n i = N (4.78) i ( ) 2 - µn H = ϕ n i n j, µ i n i (4.79) 11 66

13 N S i = 2n i 1 n i = 1 + S i 2 (4.80) (4.80) n i S i H = ϕ S i S j 4 ( zϕ 4 + µ ) S i µn 2 i 2 (4.81) J = ϕ 4, h = zϕ 4 + µ 2 (4.82) 67

14 [ (Gauss) ] π e ax2 dx = a x 2 e ax2 dx = 1 2a π a x 2n e ax2 dx = (2n 1)!! π 2 n a n a (4.83) (4.84) (4.85) [ (Stirling) ] n 1 ln n! n ln n e n! ( ) n n ( 2πn ) e 12n + (4.86) (4.87) [ ] n [d ] [3 ] [ ] Γ(z) = 0 t z 1 e t dt (4.88) Γ(n + 1) = n! (4.89) Γ(n ) = (2n)! π (4.90) 2 2n n! V d (R) = D(ε) = π d/2 (d/2)γ(d/2) Rd (4.91) gv m3/2 2 1/2 π 2 h 3 ε1/2 (4.92) λ T = ( 2π h 2 mk B T ) 1/2 = h 2πmkB T (4.93) 68

15 Sommerfeld ( F (ε) df(ε) ) dε = F (µ) + 2 C 2n (k B T ) 2n d2n F (ε) dε n=1 dε 2n (4.94) ε=µ f(ε) = 1/[exp{(ε µ)/k B T } + 1] C 2n 1 x 2n 1 dx (4.95) Γ(2n) 0 e x + 1 = ( n) ζ(2n) (4.96) Riemann ζ(m) r=1 1 r m (4.97) (4.94) F (ε)f (ε)dε = F (µ) + π2 6 (k BT ) 2 F (µ) + 7π4 360 (k BT ) 4 F (µ) + (4.98) F (ε) = ϕ(ε) 0 ϕ(ε)f(ε)dε = µ 0 ϕ(ε)dε + π2 6 (k BT ) 2 ϕ (µ) + 7π4 360 (k BT ) 4 ϕ (µ) + (4.99) 0 x n 1 z 1 e x 1 dx = Γ(n) z k k Γ(n)g n(z) (4.100) n g n (1) = ζ(n) ( 1 Γ 2) = ( 3 π, Γ = 2) k=1 ( ) π 5 2, Γ = 3 π 2 4, (4.101) ζ(2) = π2 π4 π6 = 1.645, ζ(4) = = 1.082, ζ(6) = = 1.017, ( ( 3 5 ζ = 2.612, ζ = 1.341, ζ(3) = 1.202, 2) 2) ( 7 ζ = 1.127, ζ(5) = (4.102) 2) 69

1 s 1 H(s 1 ) N s 1, s,, s N H({s 1,, s N }) = N H(s k ) k=1 Z N =Tr {s1,,s N }e βh({s 1,,s N }) =Tr s1 Tr s Tr sn e β P k H(s k) N = Tr sk e βh(s k)

1 s 1 H(s 1 ) N s 1, s,, s N H({s 1,, s N }) = N H(s k ) k=1 Z N =Tr {s1,,s N }e βh({s 1,,s N }) =Tr s1 Tr s Tr sn e β P k H(s k) N = Tr sk e βh(s k) 19 1 14 007 3 1 1 Ising 4.1................................. 4................................... 5 3 9 3.1........................ 9 3................... 9 3.3........................ 11 4 14 4.1 Legendre..............................