( ) I( ) TA: ( M) 015 7 17
, 7 ( ) I( ).., M. (hatomura@spin.phys.s.u-tokyo.ac.jp).,,.. Keywords: 1. (gas-liquid phase transition). (critical point) 3. (lattice gas model) (Ising model) H = ϕ 0 i,j n i n j µ i n i, n = 0 (for vacancy), n = 1 (for occupation). (1) σ i = ±1, n i = σ i + 1. () H = J i,j σ i σ j h i σ i, σ i = 0 (for vacancy), σ i = 1 (for occupation), (J = ϕ 0 /4, h = (zϕ 0 +µ)/4) 4. L. Onsager (1944), C. N. Yang (195) 5. (Monte Carlo simulation) 6. XY, Potts (3),,,., ( ).
3 1 5 1.1................................................. 5 1........................................ 7 1.3 (phenomenological free energy, Ginzburg-Landau (GL) ) f(m, T, H)................................................. 7 1.4 Bragg-Williams........................................... 8 1.5............................................. 9 1.6.................................... 10 15.1................................................... 15............................................ 16.3............................................. 17.4............................................ 0.5.............................................6 Berezinskii-Kosterlitz-Thouless [3]................................. 4 3 7 3.1............................................ 7 4 31 4.1............................................ 31 5 Monte Carlo 33 5.1 Monte Carlo.............................................. 33 5.,,.................................. 33 5.3,.................................. 35 6 37 6.1 015/06/01.................................... 37 39
5 1 1.1, Ising Hamiltonian : Z = Tre βh, (1.1) H = J σ i σ j H σ i, i,j σ i = ±1, (1.) σi = M. (1.3) J = 0,,, : Z = e βh σi = ( ) e βhσi = ( cosh(βh)) N, σ i =±1 i σ i =±1 (1.4) F = k B T N ln[ cosh(βh)]. (1.5) J 0, Z = e βj i,j σ iσ j +βh σ i, (1.6) σ 1=±1 σ N =±1,.,., 1.1, i, σ j σ j = m., (1.), J i,j σ iσ j H σ i (Jzm + H) σ i H (1) MF., z., z = 4. 1.1
6 1 1.,.,, m = σ i = Trσ ie βh (1) MF Tre βh(1) MF. : = σ σeβ(jzm+h)σ σe β(jzm+h)σ = tanh(βjzm + βh), (1.7) m = tanh(βjzm + βh). (1.8), H = 0 1., βjz < 1 ( ) m = 0, βjz > 1 ( ) m = 0, ±m s.,. (1.8) H = 0, m = tanh(βjzm) = βjzm 1 3 (βjzm)3 +, (1.9), m s (T c T ) 1/ = (T c T ) β, (1.10)., β = 1/, (1.11)., m = m s + χh (1.8) H, m s + χh = tanh(βjzm s ) + β(jzχ + 1)H cosh (βjzm s ) +, (1.1), χ (T T c ) 1 = (T T c ) γ, (1.13).,.. γ = 1, (1.14)
1. 7 1. H = 0. σ i = (σ i m) + m, (σ i m), H = J σ i σ j = J i m + m)(σ j m + m) i,j i,j (σ = J i,j [(σ i m)(σ j m) + m(σ j m) + m(σ i m) + m ] J i,j [m(σ i + σ j ) m ] = Jzm N σ i + JzN m H MF, (1.15).,, H 0 Z MF = Tre βh MF = Tre βjzm σ i βjzn m = [ cosh(βjzm)] N e βjzn m βjz = e { m ln[ cosh(βjzm)]}n, (1.16) F (m, T, H) = F (m) = k B T ln Z MF = Jz m k B T ln[ cosh(βjzm + βh)], (1.17). F (m, T, H), F (T, H). m, F (m) m = 0 Jzm k BT tanh(βjzm + βh)βjz = 0, (1.18), m = tanh(βjzm + βh), (1.19),. 1.3 (phenomenological free energy, Ginzburg-Landau (GL) ) f(m, T, H),., m, f(m) = am + bm 4 + cm 6 + hm, (1.0)., f(m) m = am + 4bm3 h + o(m 3 ), (1.1)., b > 0, a > 0, a < 0 ( 1.3)., a (T T c )., (1.13) 1,,
8 1 1.3 1.4 Bragg-Williams a = a 0 (T T c ) : a > 0, am h = 0, m = h a = χh, χ = 1 a = 1 a 0 (T T c ), (1.) a a < 0, a = 4bm, m = b T T c 1. (1.3) T = T c, 4bm 3 = h, m h 1 3 = h 1 δ δ = 3. 1.4 Bragg-Williams Bragg-Williams ( 1.4)., F = E T S, (1.4)., E, S. p(s) E = N Jz m HNm, (1.5) S = k B s:all states p(s) ln p(s), (1.6) N p(s) p i (σ i ), (1.7) p i (σ i ) = { pi (+) p i ( ), (1.8)
1.5 9, S = k B {σ i=±1} p 1 p p N ln p 1 p N = k B N[p(+) ln p(+) + p( ) ln p( )] { 1 + m = k B N ln 1 + m + 1 m ln 1 m }, (1.9) F = N Jz { 1 + m m HNm + k B T N ln 1 + m + 1 m ln 1 m }, (1.30). { pi (+) + p i ( ) = 1 p i (+) p i ( ) = m, (1.31),. p i (+) = 1 + m p i ( ) = 1 m, (1.3), 1.1, 1.. F (m) m = 0, (1.33), f = F/N { f 1 m = Jzm H + k BT ln 1 + m + 1 + m 1 1 + m 1 ln 1 m + 1 m } 1 1 m = Jzm H + 1 k BT ln 1 + m = 0, (1.34) 1 m. m = tanh(βjzm + βh), (1.35), Bragg-Williams.,. tanh 1 X = 1 ln 1 + X 1 X, (1.36) 1.5 1.3 1.3, 0.,., 1.5,.,, (metastable state).,,. (spinodal point)., (magnetization process) m(h),. 1.6, s.p.,., m(h) H, H m
10 1 1.5 1.6 1.7,., H m = tanh(βjzm + βh), (1.37) H(m) = k BT ln 1 + m Jzm, (1.38) 1 m, m. 1.6,, ( 1.7). i., m i = σ i = Trσ ie βj(mj 1 + +mj z )σi+βhiσi Tre βj( )σi+βhiσi, (1.39) m i = tanh[βj(m j1 + + m jz ) + βh i ], (1.40)
1.6 11. T > T c, m i 1, m i βj(m j1 + + m jz ) + βh i, (1.41), Fourier σ k = 1 N σ j = k N σ j e i k r j j=1 σ k e i k r j, (1.4), σ k e i k ri = βj σ k (e i k rj1 + + e i k rjz ) + βhi, (1.43) k k., e i k r i i, σ k = 1 βh i e i k r i + βj(e i k δ 1 + + e i k δ z ) σ k, (1.44) N i., σ k = 1 N i βh ie i k r i 1 βj, (1.45) m e i k δ m., 1 βj m e i k δ m = 1 βj z/ m=1 cos( k δ m ) = 1 βjz + βj(k x + k y + ) + = 1 βjz + βjk +, (1.46), k = k H k σ k β 1 βjz + βjk, (1.47).,. H k 1 H i e i k r i, (1.48) N i G ij = σ i σ j σ i σ j, (1.49). (1.39), σ j = βtrσ jσ i e [ ] βtrσ je [ ] Trσ i e [ ] H i Tr (Tr ) = β[ σ i σ j σ i σ j ], (1.50) G( r j ) G 0j = σ 0 σ j σ 0 σ j = σ j H 0 1 β, (1.51)
1 1. Fourier G( k) = 1 G( r j )e i k r j N j = 1 β H 0 σ k, (1.5), (1.47) G( k) = 1 N ei k 0 1 βjz + βjk = 1 1 N 1 βjz + βjk, (1.53). Fourier, G( r) = k e i k r G( k) = 1 (π) d e i k r d k 1 βjz + βjk, (1.54). k = N (π) d d k, (1.55)., 1 (π) d e i k r a + k d k = 1 r e ar, r = r, (1.56). (correlation length) ξ, r ξ Ornstein-Zernike ξ = a 1 = 1 βjz, (1.57) βj G( r) 1 e r/ξ, (1.58) r d 1 r ξ., G( r) 1 r d 1 r d e r/ξ, (1.59) ξ = 1 βjz βj T T c, (1.60), ξ (T T c ) 1 = (T Tc ) ν, ν = 1, (1.61)., η 1 G( r), rd +η η = 0 (1.6)., C (T T c ) α, α = 0, (1.63) m (T T c ) β, β = 1, (1.64) m H 1 δ, δ = 3, T = Tc, (1.65)
1.6 13 χ (T T c ) γ, γ = 1, (1.66)., GL., f(m) = am + bm 4 + hm, (1.67), m m( r)., GL ϕ 4 f(m) F [{m( r)}] = d r[( m) + am + bm 4 hm], m( r), (1.68)., 1. δf = 0, (1.69) δm,. h = 0, m + am + 4bm 3 h = 0, (1.70) m + am + 4bm 3 = 0, (1.71), a < 0, T < T c, (a T T c ),. m(x) = A tanh(x/ξ), ξ = a a, A = b, ξ a (T T c ) 1, ν = 1/, (1.7)
15.1 A, A, A A., = Tr e βh, (.1) Tre βh., H = H 0 aa, a A ( ), A.,. Z(a) = Tre βh 0+βaA, (.) A = TrAe βh Z(a) = ln Z(a), (.3) (βa) A A = TrA e βh ( ) TrAe βh = ln Z, (.4) Z Z (βa) χ AA = a A = β (βa) (βa) ln Z(a) = A A, (.5) k B T, Kirkwood., χ (m) = m A = βm m+1 am (βa) m+1 ln Z(a) = βm A m+1 c, (.6)., A m+1 c (m + 1) (cumulant)., A 3 c = A 3 3 A A + A 3, (.7)., ln Z(a) = βf O(N).,., A = M = N σ i, a = H, (.8)
16, = 1 k B T χ MM = M M k B T = 1 N k B T = 1 k B T N N j=1 N σ i j=1 N j=1 N N σ j σ i σ j j=1 N (σ i σ i )(σ j σ j ), (.9) (σ 1 σ 1 )(σ j σ j ) = 1 N k B T N C(1, j), (.10)., 1 j r 1j = r C(1, j) 1 r d +η e r/ξ, = N d d r e r/ξ k B T V r d +η N k B T V ξ η d d x e x, (.11) xd +η., 1 χ 0, χ MM = Nχ 0. j=1 χ 0 ξ η, (.1). (Husimi-Temperley ) :, Z =., = = σ 1 =±1 N M= N 1 1 ( H = Jz N ) ( N ) σ i H σ i. (.13) N σ N =±1 e βh N C N+ e β Jz N M +βhm dme β Jz N N m +βhnm N[ 1+m ln 1+m + 1 m ln 1 m ], (.14) { N+ + N = N N + N = M, N ± = N ± M, N! N N e N, m = M/N, (.15)., N T > T c O(N) M N = T = T c O(N 3/ ), (.16) T < T c O(N )
.3 17..,., Z = Tre β Jz N ( σ i ) +βh σ i 1 = Tr π = 1 π dxe x ( +x βjz N N )+βh σi N σi dxe x N cosh N (x ) βjz N + βh, (.17) x = Ny,. = N dye Ny +N ln( cosh[y βjz+βh]), (.18) π 1 π dxe x +ax = e 1 a, (.19)., K = βjz M = N Z Z K = N π Z K,., (6.). (.0) N tanh( ( y Ky) y ) K e N ln cosh( Ky) dy, H = 0, (.1),,..3,. 1 Ising : H = J H = 0. Z = Tre βh = = σ 1=±1 {τ i =±1} N 1 σ N =±1 σ i σ i+1 H N σ i. (.) e βj(σ1σ+ +σ N 1σ N ) e βj(τ1+ +τ N 1) = [ cosh βj] N 1, (.3)., σ i σ i+1 = τ i = ±1, (.4)
18.., : σ i σ j = τ i τ i+1 τ j 1 = e βjτ1 e βjτ i 1 τ i e βjτi τ j 1 e βjτ j 1 e βjτj e βjτ N 1 /Z {τ i =±1} = [ cosh βj]n 1 (j i) [ sinh βj] (j i) [ cosh βj] N 1 = (tanh βj) j i ln[tanh βj](j i) = e = e (j i)/ξ. (.5), ξ ξ = ln tanh βj, (.6)., 1 XY : H = J N 1 cos(θ i θ j ) H N cos θ i. (.7) XY Ising,. ( ), XY : H = J i,j cos(θ i θ j ) H N cos θ i E 0 + J (θ i θ j ) + H = E 0 + k i,j,., J E 0 = Jz N NH, θ k = 1 N θ i [ cos k x cos k y + H N j θ j e i k r j, θ j = 1 N ] θ k θ k, (.8) k θ k e i k rj, (.9). π < θ i < π < θ i <,, cos θ j = e iθ j = e 1 θ j (.30)
.3 19., θ j = 1 N = 1 N = 1 N = 1 N N θj j=1 N 1 θ k e i k rj θ k e i k r j N j=1 N j=1 1 N k k e i( k+ k ) r j θ k θ k k, k k θ k θ k, (.31),, 1 θ k θ k = J ( cos k x cos k y ) + H/, (.3) θj = 1 (π) πk BT (π) d k 0 J k B T ( cos k x cos k y ) + H/ 4k Jk, (.33) + H cos θ j = e 1 θ j 0 (H 0), (.34)., 1, (infrared divergence). 3, Λ 0 d d k k, (.35), S k S k < k B T J(3 cos k x cos k y cos k z ) < k BT 1, (.36) Jk 1 = 1 N N S j = 1 S k N S k j=1 k = m + 1 (π) d k S k S k, (.37) k 0 m > 1 k BT (π) 3 d k 1 k, (.38). T m > 0,., [1, ].
0.4,., T T c, ξ t = T T c T c, ξ t ν (.39). ν., b (βj). b,., ξ(t ), ξ(t ) = ξ(t )/b T, ξ ξ/b (.40) ξ = ξ/b = 1 b t ν = t ν, (.41),.,., Ising : t ν t = b, (.4) ν ν = ln b ln t, (.43) t e βh = e βj σ iσ j+βh σ i, (.44)., K = βj. Rb 1 K (K 1,, K M ) = R b (K) = = K 1.. K M. R M b (K 1,, K M ), (.45). K 1 = βj, K = βh,, K M =. (fixed point)k K = R b (K ), (.46)., ξ = 0, ( )., δk = K K, (.47)., δk = R b (δk) = K K, (.48). K,., T, δk = T δk, (.49) T ij = Ri b (K 1,, K M ) K j, (.50) K=K T u i = λ i u i, i = 1,, M, (.51)
.4 1. u i, λ i > 1 (relevant), λ i < 1 (irrelevant), λ i = 1 marginal. marginal,., λ i = b y i, y i =, ln λi ln b > 0, y i < 0. b e δl, δl 1, (.5)., u = R b (u) = T u = b y 1 1 + y 1 δl... u b ym... u, (.53) 1 + ym δl, δu = u u = y 1... uδl, (.54), δu δl = y 1... y M y M u, (.55).,. Ising : Z = e K(σ 1σ +σ σ 3 + +σ N σ 1 ). (.56) σ 1=±1 σ N =±1 K = βj., : ( ) ( ) Z = e K( ) (.57) σ 1=±1 σ =±1 = Tr cosh[k(σ 1 + σ 3 )] (.58) = Tr Ãe K (σ 1σ 3+ ). (.59), Tr, Ã K., K = 1 ln cosh(k), b = (.60). K, K 1 (T ) K = K 1 ln 1 e 4K, K 1 (T 0), (.61)., K, K K = 0.,.,.,., σ 0 : e Kσ 0(σ 1 +σ +σ 3 +σ 4 ) = Ae K 1(σ 1 σ +σ σ 3 +σ 3 σ 4 +σ 4 σ 1 )+K (σ 1 σ 3 +σ σ 4 )+K 4 σ 1 σ σ 3 σ 4. (.6) σ 0=±1
, σ i (i = 1,, 4) σ 0, K 1, K, K 4., K K 1 = 1 ln cosh(4k) 8 0 K = 1 ln cosh(4k) 8 0 K 4 = 1 8 ln cosh(4k) 1, (.63) ln cosh(k) 0 A = cosh 1/8 (4K) cosh 1/ (K).,.,., factor 4 K K = 4 1 ln cosh(4k), (.64) 8, K.,.,..5 φ 4 : H = J ij m i m j λ i (m i 1), (.65)., m(r ) = m(r) + (r r ) m(r) + 1 r r m(r) +, (.66), H = [ ] 1 Ja R ( m(r)) (λ + J)m(r) + λm(r) 4 + dr, (.67). a, : m(r) ad JR ϕ(r), (.68), H = [ 1 ( ϕ) + 1 ] ta ϕ(r) + ua d 4 ϕ(r) 4 dr, (.69). t = λ+j JR, u = λ JR. Fourier : ϕ k = drϕ(r)e ik r, ϕ(r) = 1 (π) d dkϕ k e ik r, (.70), H = 1 (π) d ta dk 1 k ϕ k ϕ k + 1 (π) d dkϕ k ϕ k + uad 4 (π) 3 dk 1 dk dk 3 ϕ k1 ϕ k ϕ k3 ϕ k1 k k 3, (.71)., 1, ϕ k ζϕ k : ϕ k ϕ k ζ ϕ k ϕ k. (.7)
.5 3 0 k Λ, b., Λ/b k Λ, Λ/b Λ., H = H 0 + g i H i (.73), Tre H = Tre H Tre H 0 i g ih i 0 Tre H 0 = Tre H 0 e g i H i 0 + 1 = Tre H 0 e g i H i 0 gi g j H i H j c + (.74)., H 0 = 1 k ϕ k ϕ k., 1 ta Λ (π) d dkϕ k ϕ k = 1 tb (a ) ( 1 (π) d Z 0 (a ) 0 0 < Dϕ < ) k e H(a, a 1 b(a ) 1 t tb., u k ) dkϕ < k ϕ< k + const. (.75) ua d 4 Λ 0 Λ Λ dk 1 dk dk 3 ϕ k1 ϕ k ϕ k3 ϕ k1 k k 3 0 (.76) 0, k 1, k, k 3 < Λ/b u ub 4 d, k 1 > Λ/b( ) u 0, k 1, k > Λ/b 0 ϕ k1 ϕ k e 1 k ϕ k ϕ k /Z 0 = (π)d k 1 δ(k 1 + k), (.77).,., ( 6ua d 4 1 Λ/b ) > Z 0 (a Dϕ < ) k ϕ < ) e H0(a k 1 ϕ < 1 k 1 k, (.78) t., Λ Λ/b 0 dk 1 1 k = a (d ) dx 1 1 1/b x a (d ) A b, (.79)., k = Λx, Λ = 1/a., { t t = b t + 1uA b b + u u = ub 4 d, (.80)., ϕ ζϕ, ζ = b η { t = b η (t + 1uA b ) u = b 4 d η u. t = u = 0 (Gauss ). t = t + δt, u = u + δu ( ) δt δu = ( b b ua b 0 b 4 d ) ( δt δu k, (.81) ), (.8). { yt = y u = 4 d, (.83), 4 > d.,, u., O(u) Gauss fixed point ( ), O(u ), (, ).
4.6 Berezinskii-Kosterlitz-Thouless [3] n =, d = ( XY )., H = J N cos(θ i θ j ) H cos θ i (.84) i,j. θ i θ j 1, H J (θi θ j ) + H θ i (.85). Fourier θ k = 1 θ j e i k r i, θ j = 1 N N j k θ k e i k rj (.86), H E g + J (4 cos(k x a x ) cos(k y a y ))θ k θ k k E g + J k θ k θ k, (.87). a ba,,,.,., cos(θ i θ j ) = S i S j = e i(θ i θ j ) = e 1 (θ i θ j ), (.88), (θ j θ 0 ) = 1 (θ j(n) θ n ) N,., g(r) = π/a 0 = k BT J n π/a 0 d k 1 e i k r (π) k = k BT πj ln r a, (r a), r = r j r 0, (.89) S 0 S r = e 1 k B ( T πj ln r r ) k B T πj a =, (.90) a d k 1 e i k r (π) k = 1 π ln r + const., r a, (.91) a., exp., (vortex).,. n : θ i = πn. (.9)
.6 Berezinskii-Kosterlitz-Thouless [3] 5, θ(r) θ(r + 1) 1 r, (.93),., E V = L 0 L ( θ) 1 d r = π rdr A ln L, (.94) 0 r S = k B ln L = k B ln L, (.95) E T S = A ln L k B T ln L = (A k B T ) ln L, (.96)., T > A/k B, T < A/k B., θ( r) = θ SW ( r) + θ V ( r), (.97), (irrotional part) ( θ SW ( r)) = 0 (Sourse-free) ( θ V ( r)) = 0 d r θ r = πn, (.98).,. θ V ( r) = πδ( r r 0 )n e z, (.99) ( θ V ( r)) = (πρ( r) e z ), (.100) θ( r) = V d r a g( r r )πρ( r) e z, (.101) g = δ( r), (.10)., βh = K d r( θ) = K d r( θ SW ) + (π) K = K d r1 d r d r a g(r) = 1 π ln r a, (.103) a ρ( r 1)ρ( r )( g( r r 1 ))( g( r r )) d r( θ SW ) d rd r πk r r >a a 4 ρ( r)ρ( r ) ln r r + πk a C r r <a d rd r a 4 ρ( r)ρ( r ) π CK { d r a ρ( r) (.104) }, r a g( r), r < a C., βh = K d r( θ SW ) πk i j n i n j ln r i r j a + πk C i n i π CK{ i n i }, (.105)
6., n i = ±1., Coulomb gas (πk = J, πk C = µ ).,., (Coulomb gas ) τ τ τ + dτ, τ = a. e βπk C ˆK, Z = ( 1 n ˆK d r (n i!) n d r 1 exp β p i p j ln r i r j {n i } D n D 1 τ., p i = πjn i. Kosterlitz [3],. ( βp βp 1 (π) (βp )( ˆKτ ) dτ τ ˆKτ ˆKτ ( 1 (βp ) dτ τ ), (.106) ), (.107) ), (.108) x = βp, (.109) y = π ˆKτ, (.110) dx ( d ln τ = (π) (βp) y ) ( ) βp = y y (.111) 4π dy d ln τ = xy (.11) x 0, βp., x y = const.., ln τ = l dl dx = 1 y = 1 x + Ct, Ct = x y, (.113), l(x) = 1 tan 1 x + const., (.114) Ct Ct, ln τ 0 τ ( ) 1 x = l(x) + l(0) = tan 1, (.115) Ct Ct. ( ( τ 1 = e l(x) l(xi) = exp tan 1 τ i Ct x i Ct tan 1 )) x, (.116) Ct, x = 0, τ ξ. tan 1 ( ) Ct x i ) ξ (tan = e 1 1 x i Ct Ct e π t α Ct = e, (.117) τ i ( ) = tan 1 Ct = π., ξ t ν, ν =., KT exp., S 0 S r r 1/4.
7 3 3.1, d d + 1.,. d,, M, : Z = Tre βĥ = φ 0 e βĥ φ 0 φ 0 = {φ} φ 0 e βĥ φ M 1 φ M 1 e βĥ φ M φ M φ 1 φ 1 e βĥ φ 0. (3.1),.,, β = M β, (3.) φ i e βĥ φ i 1 e W (φi,φi 1), (3.3) Z = {φ} e M 1 i=0 W (φi+1,φi), (3.4)., e d + 1, W (φ i+1, φ i ) φ i+1, φ i..., 1 Ising : H = J N 1 σ i σ i+1 h, 1., Z = {σ j=±1} = {σ j=±1} e βh N σ i. (3.5) e βh σ 1 e βh σ 1+βJσ 1 σ + βh σ e βh σ +βjσ σ 3 + βh σ3 e βh σ N 1+βJσ N 1 σ N + βh σ N e βh σ N, (3.6)
8 3,, Z = {σ j =±1} e βh σ i+βjσ i σ i+1 + βh σ i+1 σ i ˆT σ i+1, (3.7) e βh σ 1 σ 1 ˆT σ σ ˆT σ 3 σ N 1 ˆT σ N e βh σ N, (3.8). ˆT,., Z = σ1 σ 1 ˆT N 1 σ N e βh σ N, (3.9) σ 1,σ N e βh.,, Z PBC = σ 1 ˆT N σ 1 = Tr ˆT N, (3.10) σ 1 =±1., σ = +1 = ( ) 1, σ = 1 = 0 ( e β(j+h) e ˆT = βj e βj e β(j+h) ( ) 0, (3.11) 1 ), (3.1). ( ) λ1 0 ˆT = U U, (λ 0 λ 1 > λ ), (3.13), Z = σ,σ =±1 ( ) e βh (σ+σ ) λ N 1 σ U 1 0 0 λ N 1 U σ, (3.14)., N λ N 1 1 λ N 1, c Z cλ N 1 1., f = k BT N ln Z k BT ln λ 1, (3.15)., N., Pauli : ( ) σ x 0 1 =, σ y = 1 0 ( ) ( ) 0 i, σ z 1 0 =. (3.16) i 0 0 1,.., σ ˆT σ = e βh σ e βjσσ e βh σ σ ˆT 1 ˆT ˆT3 σ, (3.17) ˆT 1 = ˆT 3 = e βh σz, ˆT = ( e βj e βj e βj e βj ), (3.18) tanh K = e βj, (3.19) 1 ˆT = sinh K cosh K e Kσ x, (3.0)
3.1 9. ˆT = ˆT 1 ˆT ˆT3 = 1 sinh K cosh K e βh σz e Kσ x e βh σz, (3.1)., z x., ˆT,., h = 0 :, 1 h = 0, ˆT = sinh K cosh K e Kσ x. (3.) x = 1 ( + ), λ 1 = x = 1 ( ), λ =.,. σ i σ j = 1 σ i σ j e βh, (i < j) Z PBC {σ} = 1 Z PBC Tr ˆT i 1 σ z ˆT j i σ z ˆT N j+1 e K sinh K cosh K e K sinh K cosh K, (3.3), (3.4) = 1 Z PBC {λ N+i j 1 x σ z ˆT j i σ z x + λ N+i j x σ z ˆT j i σ z x }, (3.5) σ z x = x, σ z x = x, (3.6),. Z PBC λ N 1 σ i σ j = 1 Z PBC {λ N+i j 1 λ j i + λ N+i j λ j i 1 }, (3.7) σ i σ j ( λ1 λ ) i j = exp( i j /ξ), ξ 1 = ln λ 1 λ, (3.8).,.,,. σ i = ( λ λ 1 ) i 1 ( λ λ 1 ) N i = e (i 1)/ξ e (N i)/ξ, (3.9) Ising., : H = J j (σ 1j σ 1,j+1 + σ j σ,j+1 ) J j σ 1j σ j. (3.30), 4 4 ˆT = e K+K 1 1 e K+K 1 e K K e K K 1 1 e K K e K K 1 e K+K 1 1 e K+K, K = βj, K = βj, (3.31)
30 3., { + 1, +1, + 1, 1, 1, +1, 1, 1 }, (3.3)., 4 4., Perron-Frobenius,.,., σ 1 σ ˆT σ 1σ = e K σ1σ+k(σ1σ 1 +σσ ) e K σ 1 σ, (3.33) 1 ˆT = sinh K cosh K e K σz 1 σz x e K(σ 1 +σx ) e K σz 1 σz (3.34)., z x. M N., M Ising. 1 Perron-Frobenius d d A; 1. i, j A ij 0, A ij R. n (A n ) ij > 0,,. λ 1 λ 1 > λ i, i.
31 4 4.1, F = E T S, (4.1) E S., 0,.., S(0)/Nk B = 0.3383, Villain S(0)/Nk B = 0.916, S(0)/Nk B = 0.50.,,,,.,., H = J nn σ i σ j J nnn σ i σ k, (4.) i,j i,k., nn (nearest neighbor), nnn (next nearest neighbor),., m A = σ A = tanh( βj nn (3m B + 3m C ) + 6J nnn m A + βh) m B = σ B = tanh( βj nn (3m C + 3m A ) + 6J nnn m B + βh), (4.3) m C = σ C = tanh( βj nn (3m A + 3m B ) + 6J nnn m C + βh)., m A = m B = m C disordered phase, m A = m B, m C = 0 partially disordered phase, 3-sublattice phase, -sublattice phase..
33 5 Monte Carlo 5.1 Monte Carlo Z = e βei, (5.1) all states {i},., i., A = e βe i Z, (5.) B = sample sample E i e βe i E i Z, (5.3), B/A E, (5.4).., N.,,., i e βe i., E i sampling sampling 1 = E i, (5.5)..,. 5.,, {w i j }., i p i, i = 1,, M,, p = (p 1,, p M )., 1 p = (p +, p ), { p+ (t + t) = p + (t)(1 w + t) + p (t)w + t p (t + t) = p + (t)w + t + p (t)(1 w + t), (5.6)
34 5 Monte Carlo. Master equation, {w i j } Markov., p(t + t) = L( t) p(t), (5.7) L( t)., L( t) L ij = w j i t, for i j L ii = 1 w i j t, (5.8) j i., i L ij = 1., L p, p(t + n t) = (L( t)) n p(t), p(t) = p(t), (5.9). L( t) ϕ i = λ i ϕ i, (5.10), p(0) = i c i ϕ i, (5.11) M M p(n t) = L n p(0) = L n c i ϕ i = c i λ n i ϕ i c max λ n max ϕ max, for large n, (5.1). λ max L. ( λi ) n 0, as n, (5.13) λ max., λ max.., L ij = w j i t 0 =, ; n (L n ) ij > 0., (i j)., n j=1 p j(t) = 1 n j=1 p j(t + t) = 1, p(0) p(t) = p(n t) = L n p(0) = p stationary, (5.14) λ max = 1, c max = 1., M j=1 ϕ (j) max = 1, M j=1 ϕ (j) k max = 0, (5.15)., p(n t) = p st + M c i λ n i ϕ i = p st + M c i e n ln λi ϕ i, p st = ϕ max, (5.16), max, max., ϕ i i, c i, τ i = 1/ ln λ i.,., p st = p eq L., p eq = L p eq = (p 1 eq,, p M eq), (5.17)
5.3, 35, p (m) eq = p (m) eq (1 w m l t) + p (l) eq w l m t, (5.18) l m l m. 0 = (w m l tp (m) eq l m w l m tp (l) eq ), p (m) eq = 1 Z e βem, (5.19), detailed balance w m l p (m) eq = w l m p (l) eq, (5.0)., (Glauber ) Metropolis., p(t), p(t) t.. = L p(t) (5.1) 5.3,, p(i) i e βh i = m i E m E m i e βem (5.), i e βh i = i e βh/m e βh/m e βh/m i = i e βh/m j 1 j 1 j M 1 e βh/m i, (5.3) j 1,,j M 1. M e βh/m 1 β H, (5.4) M, j k e βh/m j k+1 j k (1 βm ) ( H j k+1 = δ jk j k+1 β ) M j k H j k+1, (5.5).,., d, (d + 1).,. W jp j p W j p j p β = j p 1 e M H j p j p e β M H j p+1 j p 1 e β M H j p j p e β M H j p+1, (5.6),., N H = J σi z σi+1 z Γσi x, (5.7)
36 5 Monte Carlo., J > Γ (T = 0 σ z = 0), J < Γ (T = 0 σ z = 0)., ( ) e β M H e β M (J σ z i σz i+1 ) e β M Γ σ x β + o, (5.8) M. Suzuki-Trotter., e A+B = (e A/n e B/n ) n, as n, (5.9) Trotter., σ i e βγ M σx i σ i = Ae KSTσiσ i, (5.30),, e βγ M σx = cosh βγ M + σx sinh βγ M, (5.31) cosh βγ M = AeKST sinh βγ, (5.3) M = Ae K ST A = cosh βγ M K ST = 1 sinh βγ M ln tanh βγ M, (5.33).,,., 1 Heisenberg., H = J N σ i σ i+1, (5.34) e βh = e βj(σ1σ+σσ3+ +σ N σ 1) = e βj[(σ 1σ +σ 3 σ 4 + )+(σ σ 3 +σ 4 σ 5 + )], (5.35) Suzuki-Trotter.,, negative sign problem.
37 6 6.1 015/06/01 1. (Husimi-Temperley ) : ( H = Jz N ) ( N ) σ i H σ i. (6.1) N N T > T c O(N) M N = T = T c O(N 3/ ), (6.) T < T c O(N )... : α + β + γ = γ = β(δ 1) γ = ( η)ν α + β(1 + δ) = α = dν Rushbrook Widom Fisher Griffis Josephson, (6.3) : f(u) = b d f(u ) = b d f(b y1 u 1, b y u, ), (6.4).
39 [1] J. Frölich et al., Phys. Rev. Lett. 36, 804 (1976). [] F. J. Dyson et al., J. Stat. Phys. 18, 335 (1978). [3] J. M. Kosterlitz, J. Phys. C 7, 1046 (1974).