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1 2016 Kosterlitz-Thouless Haldane Dept. of Phys., Kyushu Univ
2 2016 Figure: D.J.Thouless F D.M.Haldane J.M.Kosterlitz TOPOLOGICAL PHASE TRANSITIONS AND TOPOLOGICAL PHASES OF MATTER
3 ( ) ( ) (Dirac, t Hooft-Polyakov)
4 BKT J. M. Kosterlitz, and D.J. Thouless: Journal of Physics C 6 p (1973). 2 XY 2 U(1) XY O(2) ) 2 (Mermin-Wagner (1966)) But, 2 XY 2 ( ( ) Berezinskii: Sov. Phys. JETP, 32, p.493 (1971); Sov. Phys. JETP, 34, p.610 (1972)
5 BKT Kosterlitz (1973) Kosterlitz-Thouless (1973) Kosterlitz (1974) J. M. Kosterlitz: J. Phys. C, 7, pp (1974). Anderson-Yuval :Phys. Rev. Lett, 23, (1969) 89 ; Anderson-Yuval-Hamann :Phys. Rev. B, 1, (1970)
6 Haldane F.D.M. Haldane: Phys. Letters A, 93, p.464 (1983); Physical Review Letters, 50, p.1153 (1983) S=1/2 1 ) ( )
7 TKNN D. J. Thouless, Mahito Kohmoto, M.P. Nightingale, and M Den Nijs: Physical Review Letters, 49(6):405, (1982) ( )
8 (winding number) Figure: : -2, : -1, : 0 Figure: :1, : 2, : 3
9 ( ) 3.1 ( 1 ( x 2π C r 2 dy y ) r 2 dx 3.2 ( ) 1 2πi dz z a C (1) (2) 3.3 ( ) ( )
10 2 XY XY : H XY = J <i,j> S i S j = J <i,j> cos(θ i θ j ) (3) ( θ j (0 θ j < 2π) S j (cos θ j, sin θ j ). < i, j > )
11 2 XY 2 XY
12 2 XY (vortex) (anti-vortex) 1. 1 XY 2. 1 XY 3. ( 4.
13 2 XY BKT
14 2 XY BKT ( ) ( )
15 BKT 2 1. c 1 (T < T c ) S(r) S(r ) c 2 r r η (T = T c ) c 3 exp( r r /ξ(t )) (T > T c ) (4) T c 2. 2 XY S(r) S(r ) { c 1 r r η(t ) (T T c ) c 2 exp( r r /ξ(t )) (T > T c ) (5)
16 (3) cos(θ i θ j ) 1 (θ i θ j ) 2 /2 1 ( θ) 2 /2. H = E 0 + J d 2 r( θ(r)) 2 (6) 2 (E 0 = 2JN ) (β = 1/(k B T )) ( Z = exp( βe 0 ) D[θ] exp β J ) d 2 r( θ(r)) 2 (7) 2 Green 2 ln(r) = 2πδ(r) Γ(r r) θ(r )θ(r) = 1 2π ln r r (8)
17 2 ( ) 1. 1 W (h) = dθ exp( 1 2 Aθ2 + ihθ) = (2π/A) 1/2 exp( 1 2 A 1 h 2 ) (9) exp(ihθ) W (h) W (0) = exp( 1 2 A 1 h 2 ) (10) 2. N exp( 1 2 N i=1 N j=1 θ ia i,j θ j ) exp(i i h i θ i ) = exp 1 2 N i=1 j=1 N h i (A 1 ) i,j h j (11)
18 3 ( ) 3. exp( 1 2 θ(r)a(r, r )θ(r )d d rd d r ) W (h) exp(i h(r)θ(r)d d r) ( = exp 1 ) h(r)a 1 (r, r )h(r )d d rd d r 2 (12) A 1 A 1 (r, r )A(r, r )d d r = δ d (r r ) (13) 4. θ(r )θ(r) = δ2 W (h) δh(r 1 )δh(r 2 ) = A 1 (r 1, r 2 ) (14)
19 4 XY S(r) S(r ) = exp(i(θ(r) θ(r ))) = exp( k BT 2πJ Γ(r r )) ( ) a kb T/2πJ = r r (15)
20 < θ < θ 1
21 < θ < θ 1 But, θ = θ + 2π ( ) v v 1 2π C dl θ(r) (16) S(r)(= exp(iθ(r))) v = n (n: )
22 2πn = dl θ(r) = 2πr θ (17) θ = n/r θ(r = n/r E vor E 0 = J d 2 r( θ(r)) 2 2 C = Jn2 2 2π 0 dθ L a rdr ( ) 1 2 = Jπn 2 ln L r a (18) (L a )
23 ( +1) ( -1) ( 0 ( r 2πJ ln a) (19) (r )
24 ( ) ( ) L L 2 F = E T S Jπ ln k B T ln a a 2 (20) T KT Jπ/(2k B )
25 S = 1 ( µ θ) 2 d 2 x, (θ θ + 2π) (21) 2g θ θ sw θ vortex θ(x) = θ sw (x) + θ vortex (x) (22) dθ sw (x) = 0, (23) dθ vortex (x) = v (v : ) (24) ψ ( ) ϵ µν ν ψ = µ θ vortex (25)
26 v = dθ vortex (x) = 1 2π 2 ψd 2 x (26) 2 ψ = 2π j v j δ(x x j ) (v j : ) (27) 2 Green 1 2π ln x ψ(x) = 2π j v j 1 2π ln x x j = j v j ln x x j (28) θ vortex (x) = Im j v j ln(x x j ) (29) ( )
27 2D Coulomb gas Action S vortex = 2π 2g v i v j ln z i z j (30) i,j (2 ( )) Action sine-gordon ( < χ < ) L = 1 2g ( χ)2 2 cos( 2πχ g ) (31)
28 2 sine-gordon ( ) α α = α exp(dl) α(1 + dl) L l 0 = ln L (1/ ln L) dy 1 (l) = y 2 dl 2(l) dy 2 (l) = y 1 (l)y 2 (l) (32) dl
29 : K. Nomura: J. Phys. A, Vol. 28, pp (1995); Nomura and A. Kitazawa: J. Phys. A: Vol. 31 (1998) pp.7341
30
31 Haldane 1 H = J j S j S j+1 (33) Néel (S=1/2,3/2,...) (S=1,2,...) 0 π
32 Haldane : 1 2g dtdx ( 1 v ( ) φ 2 v t ( ) ) φ 2 x (φ (φ 1, φ 2, φ 3 ), φ 2 = 1) (34) Wick 1 2g dx 2 ( φ) 2 (35) 2 )
33 Haldane : 1 2g dtdx ( 1 v ( ) φ 2 v t ( ) ) φ 2 x (φ (φ 1, φ 2, φ 3 ), φ 2 = 1) (34) Wick 1 2g dx 2 ( φ) 2 (35) 2 ) massless(gapless)
34 Haldane : 1 2g dtdx ( 1 v ( ) φ 2 v t ( ) ) φ 2 x (φ (φ 1, φ 2, φ 3 ), φ 2 = 1) (34) Wick 1 2g dx 2 ( φ) 2 (35) 2 ) massless(gapless) But φ 2 = 1
35 Haldane : 2 ( ) φ
36 Haldane : 2 ( ) φ Figure: ( ) gap
37 Haldane : 2 ( ) φ Figure: ( ) gap 4 Yang-Mills ( ) 2
38 Haldane :, 0 π ˆφ 2i 1 2s (Ŝ2i+1 Ŝ2i) a 0, s ˆl 2i 1 2a (Ŝ2i+1 + Ŝ2i) (36) [ˆl a (x), ˆl b (y)] = iϵ abcˆlc δ(x y) [ˆl a (x), ˆφ b (y)] = iϵ abc ˆφ c δ(x y) [ ˆφ a (x), ˆφ b (y)] = iϵ abcˆlc a2 δ(x y) 0 (37) s2 δ(x y) = lim a 0 δ x,y a
39 Haldane : 2 ˆφ 2i ˆl 2i = 1 2s (Ŝ2i+1 Ŝ2i) 1 2a (Ŝ2i+1 + Ŝ2i) = 1 2sa (Ŝ2 2i+1 + Ŝ2i+1 Ŝ2i Ŝ2i+1 Ŝ2i Ŝ2 2i) = 0 ˆφ (38) ( ˆφ 2i ) 2 = 1 4s 2 [Ŝ2 2i + Ŝ2 2i+1 2Ŝ2i Ŝ2i+1] = 1 4s 2 [2Ŝ2 2i + 2Ŝ2 2i+1 4a 2 (ˆl 2i )) 2 ] = 1 + 1/s a 2ˆl/s 2 (39)
40 Haldane : 3 Ŝ 2i Ŝ2i+1 = 2a 2ˆl2 2i + s(s + 1) (40) Ŝ 2i 1 Ŝ2i = s 2 ˆφ 2i 2 ˆφ 2i as [ˆl2i 2 ˆφ 2i ˆφ 2i 2 ˆl ] 2i + a 2ˆl2i 2 ˆl 2i a 2 ( 2s 2 ( ˆφ ) 2 2s(ˆl ˆφ + ˆφ ˆl) + 2ˆl 2) s(s + 1) 2a 2 s 2 ( ˆφ ˆφ ) (41) ( ˆφ 2 = 1 + 1/s aˆl 2 /s 2 ˆφ ˆφ = ( ˆφ ˆφ) ( ˆφ ) 2 )
41 Haldane : 4 Ĥ = aj 2 dx[4ˆl 2 + 2s 2 ( ˆφ ) 2 2s(ˆl ˆφ + ˆφ ˆl)] (42) Ĥ = dxĥ, (43) [ Ĥ = v ( g ˆl θ ) ] 2 2 4π ˆφ + ( ˆφ ) 2 (44) g v = 2Jas, g = 2/s, θ = 2πs
42 Haldane : 5 Hamiltonian (44) Lagrangian L = 1 2g µ ˆφ µ ˆφ + θ 8π ϵµν ˆφ ( µ ˆφ ν ˆφ) = 1 2g ( 0φ 0 φ 1 φ 1 φ) + θ 4π φ ( 0φ 1 φ) (45) ( v = 1 [ ] Π ( ) L ( 0 φ) = 1 g 0φ + θ 4π ( 1φ φ) (46)
43 Haldane : 6 H 0 φ Π L = g ( φ Π θ ) 2 2 4π ( 1φ) + 1 2g ( 1φ) 2 (47) l φ Π Ĥ = v 2 [ ( dx g ˆl θ ) ] 2 4π ˆφ + ( ˆφ ) 2 g Q.E.D.
44 Haldane : 7 Lagrangin Wick Euclid L = 1 2g µ ˆφ µ ˆφ + iθ 8π ϵµν ˆφ ( µ ˆφ ν ˆφ) = 1 2g ( 0φ 0 φ 1 φ 1 φ) + θ 4π φ ( 0φ 1 φ) (48) ( ) ( Z = Dφ exp ) d 2 xl (49)
45 Haldane : (S 0 ) Q = 1 d 2 xϵ µν ˆφ ( µ ˆφ ν ˆφ) (50) 8π ( exp(s 0 + iθq) θ = 2πs gapped gapless
46 Haldane : 2. φ = (sin α cos β, sin α sin β, cos α) Q = 1 d 2 xϵ µν ˆφ ( µ ˆφ ν ˆφ) 8π = 1 d 2 x sin αϵ µν µ α ν β 4π = 1 D(α, β) sin α 4π D(x 0, x 1 ) dx 0dx 1 = 1 ds int (51) 4π ( D(α,β) D(x 0,x 1 ) ) : Q
47 Q µ φ + ϵ µν (φ ν φ) = 0 (52) φ 1 = 1 w ( w = φ 1 + iφ φ 3 (53) Figure:
48 2 (52) z w = 0, (z = x 0 + ix 1 ) (54) w(z) = Q j=1 z a j z b j (55)
49 Haldane : 1. S=1/2 Bethe 2. (S=3/2,5/2, ) Lieb-Schultz-Mattis I.Affleck, E.H. Lieb: Lett. Math. Phys, p. 57 (1986)
50 Haldane : Haldane AKLT(Affleck-Kennedy-Lieb-Tasaki) I. Affleck, T. Kennedy, E.H.Lieb, H.Tasaki: Physical Review Letters. 59, p.799(1987).
51 BKT 1.2 Haldane ( ) 1.3 TKNN(Thouless-Kohmoto-Nightingale-denNijs) 2.?(Umklapp )? SPTP
52 2 1., 2. ( ) 3.
N cos s s cos ψ e e e e 3 3 e e 3 e 3 e
3 3 5 5 5 3 3 7 5 33 5 33 9 5 8 > e > f U f U u u > u ue u e u ue u ue u e u e u u e u u e u N cos s s cos ψ e e e e 3 3 e e 3 e 3 e 3 > A A > A E A f A A f A [ ] f A A e > > A e[ ] > f A E A < < f ; >
2 G(k) e ikx = (ik) n x n n! n=0 (k ) ( ) X n = ( i) n n k n G(k) k=0 F (k) ln G(k) = ln e ikx n κ n F (k) = F (k) (ik) n n= n! κ n κ n = ( i) n n k n
. X {x, x 2, x 3,... x n } X X {, 2, 3, 4, 5, 6} X x i P i. 0 P i 2. n P i = 3. P (i ω) = i ω P i P 3 {x, x 2, x 3,... x n } ω P i = 6 X f(x) f(x) X n n f(x i )P i n x n i P i X n 2 G(k) e ikx = (ik) n
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2 1 (7a)(7b) λ i( w w ) + [ w + w ] 1 + w w l 2 0 Re(γ) α (7a)(7b) 2 γ 0, ( w) 2 1, w 1 γ (1) l µ, λ j γ l 2 0 Re(γ) α, λ w + w i( w w ) 1 + w w γ γ 1 w 1 r [x2 + y 2 + z 2 ] 1/2 ( w) 2 x2 + y 2 + z 2
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. 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.
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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ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 +
2.6 2.6.1 ω 0 m(ẍ + γẋ + ω0x) 2 = ee (2.118) e iωt x = e 1 m ω0 2 E(ω). (2.119) ω2 iωγ Z N P(ω) = χ(ω)e = exzn (2.120) ϵ = ϵ 0 (1 + χ) ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.121) Z ω ω j γ j f j
φ 4 Minimal subtraction scheme 2-loop ε 2008 (University of Tokyo) (Atsuo Kuniba) version 21/Apr/ Formulas Γ( n + ɛ) = ( 1)n (1 n! ɛ + ψ(n + 1)
φ 4 Minimal subtraction scheme 2-loop ε 28 University of Tokyo Atsuo Kuniba version 2/Apr/28 Formulas Γ n + ɛ = n n! ɛ + ψn + + Oɛ n =,, 2, ψn + = + 2 + + γ, 2 n ψ = γ =.5772... Euler const, log + ax x
25 7 18 1 1 1.1 v.s............................. 1 1.1.1.................................. 1 1.1.2................................. 1 1.1.3.................................. 3 1.2................... 3
i E B Maxwell Maxwell Newton Newton Schrödinger Newton Maxwell Kepler Maxwell Maxwell B H B ii Newton i 1 1.1.......................... 1 1.2 Coulomb.......................... 2 1.3.........................
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0. Intro ( K CohFT etc CohFT 5.IKKT 6.
E-mail: [email protected] 0. Intro ( K 1. 2. CohFT etc 3. 4. CohFT 5.IKKT 6. 1 µ, ν : d (x 0,x 1,,x d 1 ) t = x 0 ( t τ ) x i i, j, :, α, β, SO(D) ( x µ g µν x µ µ g µν x ν (1) g µν g µν vector x µ,y
Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x
7 7.1 7.1.1 Einstein 1905 Lorentz Maxwell c E p E 2 (pc) 2 = m 2 c 4 (7.1) m E ( ) E p µ =(p 0,p 1,p 2,p 3 )=(p 0, p )= c, p (7.2) x µ =(x 0,x 1,x 2,x 3 )=(x 0, x )=(ct, x ) (7.3) E/c ct K = E mc 2 (7.4)
N/m f x x L dl U 1 du = T ds pdv + fdl (2.1)
23 2 2.1 10 5 6 N/m 2 2.1.1 f x x L dl U 1 du = T ds pdv + fdl (2.1) 24 2 dv = 0 dl ( ) U f = T L p,t ( ) S L p,t (2.2) 2 ( ) ( ) S f = L T p,t p,l (2.3) ( ) U f = L p,t + T ( ) f T p,l (2.4) 1 f e ( U/
量子力学 問題
3 : 203 : 0. H = 0 0 2 6 0 () = 6, 2 = 2, 3 = 3 3 H 6 2 3 ϵ,2,3 (2) ψ = (, 2, 3 ) ψ Hψ H (3) P i = i i P P 2 = P 2 P 3 = P 3 P = O, P 2 i = P i (4) P + P 2 + P 3 = E 3 (5) i ϵ ip i H 0 0 (6) R = 0 0 [H,
50 2 I SI MKSA r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq
49 2 I II 2.1 3 e e = 1.602 10 19 A s (2.1 50 2 I SI MKSA 2.1.1 r q r q F F = 1 qq 4πε 0 r r 2 r r r r (2.2 ε 0 = 1 c 2 µ 0 c = 3 10 8 m/s q 2.1 r q' F r = 0 µ 0 = 4π 10 7 N/A 2 k = 1/(4πε 0 qq F = k r
I ( ) 1 de Broglie 1 (de Broglie) p λ k h Planck ( Js) p = h λ = k (1) h 2π : Dirac k B Boltzmann ( J/K) T U = 3 2 k BT
I (008 4 0 de Broglie (de Broglie p λ k h Planck ( 6.63 0 34 Js p = h λ = k ( h π : Dirac k B Boltzmann (.38 0 3 J/K T U = 3 k BT ( = λ m k B T h m = 0.067m 0 m 0 = 9. 0 3 kg GaAs( a T = 300 K 3 fg 07345
m(ẍ + γẋ + ω 0 x) = ee (2.118) e iωt P(ω) = χ(ω)e = ex = e2 E(ω) m ω0 2 ω2 iωγ (2.119) Z N ϵ(ω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j (2.120)
2.6 2.6.1 mẍ + γẋ + ω 0 x) = ee 2.118) e iωt Pω) = χω)e = ex = e2 Eω) m ω0 2 ω2 iωγ 2.119) Z N ϵω) ϵ 0 = 1 + Ne2 m j f j ω 2 j ω2 iωγ j 2.120) Z ω ω j γ j f j f j f j sum j f j = Z 2.120 ω ω j, γ ϵω) ϵ
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
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