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1 Nov., keiichi.r.ito@gmail.com, ito@kurims.kyoto-u.ac.jp ( ) Nov., / 45

2 Clay Institute.1 Construction of 4D YM Field Theory (Jaffe, Witten) Jaffe, Balaban (1980).2 Solution of Navier-Stokes Equation (Feffermann) Sinai (2005) ( ) Nov., / 45

3 .1 Boltzmann (Erdös, Yau ).2 Pauli-Fierz (Semi-Classical QED).3 ( σ model Ricci, Perelman s theory,) ( ) Nov., / 45

4 .1..2, ( ).3.4 ( ) Nov., / 45

5 =,,.1, (x, p),,.2,,..3, ( ) Nov., / 45

6 .1 : KPZ, Wilson.2 :. ( ) Nov., / 45

7 PKZ, h(x, t) KPZ eq.. Ξ white noise: t h = 2 x h + V ( x h) + Ξ V ( x h) = ( x h) 2 C ε E(Ξ(x, t)ξ(y, s)) = δ(x y)δ(s t) V ( x h). C ε. ( ): h = G (V + Ξ) G = ( t 2 x ) 1 = (4πt) 1/2 exp[ (x y) 2 /2t] ( ) Nov., / 45

8 PKZ Spohn h 3 u = (u 1, u 2, u 3 ) t u = 2 x u + V (u) + Ξ V = ( x u, M x u) C R 3, Kupiainen Wilson M = (M (1), M (2), M (3) ), M (i) = 3 3 C R 3 = u = G (V (u) + Ξ) + e t u 0 G(x, t) = e t = 1 4πt exp[ x 2 /2t] G 0 < t < ε 2. χ(t) [0, 1] u = G ε (V (u) + Ξ) + e t u 0 G ε (x, t) = e t (x, 0)(1 χ(ε 2 t)) C = C ε ( ) Nov., / 45

9 PKZ Kupiainen Wilson C ε > 0, Ξ, t(ξ) > 0 ε > 0 u ε (t, x) t [0, t(ξ)] u D ([0, t(ξ)] T ). u χ.. C ε = m 1 ε 1 + m 2 log ε 1 + m 3 ( ) Nov., / 45

10 PKZ Block Spin s ε t ε 2 t,x εx. x = ε x = 1, x = 1 x = ε 1 = L N : v (ε) (φ) = ε 1/2 ( x φ, M x φ) + ε 3/2 C ε φ = G 1 (v ε (φ) + ξ) :.1 G 1 = G L 2 + (G 1 G L 2) x > L 1 < x < L.2 (G 1 G L 2) G L 2. s s L 1, G L 2 G 1 ( ) Nov., / 45

11 PKZ ε 1, 1 ε 1 = L N, N eps 1 1 L L^2 1/eps=L^N 1 L L^(N-1) ( ) Nov., / 45

12 PKZ φ = φ 1 + φ 2 φ 1 = G L 2 (v ε (φ 1 + φ 2 ) + ξ) φ 2 = (G 1 G L 2) (v ε (φ 1 + φ 2 ) + ξ) G sf (t, x) = L 1/2 f (t/l 2, x/l) ( s 1 f (t, x) = L 1/2 f (L 2 t, Lx) ) G L 2 G 1 : φ 1 = sφ, φ 2 = sζ ( ) Nov., / 45

13 PKZ φ = s 1 G L 2 (v ε (s(φ + ζ)) + ξ) = G 1 (Sv ε (φ + ζ) + ξ) = G 1 (Sv ε (φ + ζ(φ )) +ξ) }{{} Rv ε (φ ) = G 1 (Rv ε (φ ) + ξ) ζ = Γ (Sv ε (φ + ζ) + ξ) Γ = G 1/L 2 G 1 = e t (x, 0)(χ(t) χ(l 2 t)) (Sv)(ϕ) = L 2 s 1 v(sϕ) v = ( x φ) k (k > 3) S( x φ) k = L (3 k)/2 ( x φ) k ( ) Nov., / 45

14 PKZ ζ = Γ (Sv ε (φ + ζ) + ξ) Γ Sv ε (φ ) + Γ ξ φ = G 1 (Sv ε {φ + θ + Γ (Sv ε (φ )} + ξ) G 1 (Sv ε (φ + θ)) = L 1/2 ε 1/2 ( x (φ + θ), M x (φ + θ)) + L 3/2 ε 3/2 C ε θ = Γ ξ ( ) Nov., / 45

15 PKZ, ( x φ) k. Γ = e t (χ(t) χ(l 2 t)) 1 L 1, ζ φ + ζ = φ + ζ(φ ),. Γ. ζ θ (Γ ξ), ε = L N N ( x φ + θ, M( x φ + θ)) C C E(θ(t, x)θ(s, y)) = m 1 L N + m 2 N + O(1) ( ). ( ) Nov., / 45

16 PKZ 4D, 2D, NV.1,..2, (relevant ) ( ) Nov., / 45

17 .1 :, R.Peierls (1936), L.Onsager (1944).2 XY (O(2) ) Kosterlitz-Thouless (Nobel prize 2016) J.Fröhlich and T.Spencer (1982).3 2 (SO(N) ) ( ) ( ) ( ) Nov., / 45

18 [ ] Peierls : +1, 1. {s x ±1} exp[β nn (s i s j 1)] = exp[ 2β i γ i ], γ 0 {0} s 0 1 γ : Prob(s 0 = 1) < γ 0 exp[ 2β γ ] < 1/2, β 1 s 0 > 0 [ ], Ising.! Peierls 85! ( ) Nov., / 45

19 [ ] γ: : s i = 1, : s i = +1 [ O(N) ] flip-flop. ( ) Nov., / 45

20 2 2 =Landau-Ginzbrg(2 ) f (ϕ) = f (ϕ) exp[ W 0 (ϕ)] x d N ϕ(x) W 0 = 1 2 ϕ, ( + m2 0 )ϕ + g 0 2N : ϕ2 : G, : ϕ 2 : G N : ϕ 2 : G (x) = (x) NG(0), β = G(0) i=1 ( ) xy = 4δ xy δ 1, x y, Lattice Laplacian G(0) = β m0 2 32e 4πβ : ϕ 2 i G(x) = 1 + m0 2 (x) = π π π π e ipx m dpi (1 cos p i ) 2π ( ) Nov., / 45

21 2 G(0) = β m e 4πβ : G(x) = = 1 + m0 2 (x) π π π π e ipx m (1 cos p i ) dpi 2π exp[ m 0 x ], m β,. ( ) Nov., / 45

22 2 ) 1 G 0 (x, y) = + m0 2 (x, y) G n (x, y) = 1 L 4 1 x y d 2 ζ,ξ 0 G n 1 (Lx + ζ, Ly + ξ) ϕ n (x) = (Cϕ n 1 )(x) = 1 L 2 C = = (L 2 ) + (Lx x): ζ 0 ϕ n 1 (Lx + ζ) ( ) Nov., / 45

23 2 : ϕ n (x)ϕ n (y) = G n (x, y) ϕ n (x) = A n+1 ϕ }{{ n+1 (x) + z } n (x) }{{} ϕ n, Gn 1 ϕ n Λn = ϕ n+1, G 1 n+1 ϕ n+1 Λn+1 + z n, Gn 1 z n Λn A n+1 z n Λ n = L n Λ CA n+1 = 1, Cz n = 0 A n+1 = G n C + G 1 n+1 : RΛ n+1 R Λn ( ) Nov., / 45

24 2 Q : R Λ n R Λ n, CQ = 0 Q + Gn 1 Q = Γ 1 n z n = QΓ 1/2 n ξ ϕ, ( + m0 2)ϕ {z n = QΓ 1/2 n ξ n }. ϕ, G 1 0 ϕ = ϕ 1, G 1 1 ϕ 1 + ξ 0, Q + G 1 0 }{{ Q ξ 0 } Γ 1 0 = ϕ 2, G 1 2 ϕ 2 Λ2 + ξ 1, Q + G 1 1 }{{ Q ξ 1 Λ1 + ξ 0, Q + G 1 0 } Q ξ 0 Λ0 }{{} Γ 1 1 Γ 1 0 Γ n = Q + Gn 1 Q : ( ) Nov., / 45

25 2 :. L n m 1 m. exp[ W n+1 (ϕ n+1 )] = exp[ W n (A n+1 ϕ n+1 + z n )]dµ 0 (ξ) W 0 = 1 2 ϕ 0, ( )ϕ 0 + g 0 2N : ϕ2 0 :, : ϕ2 0 : z 0 = QΓ 1/2 0 ξ, ξ N(0, 1) Γ 0 = (Q + ( )Q) 1 ( ) Nov., / 45

26 : 2 ϕ n+1 : ϕ 2 n+1 : D : ϕ n+1(x)ϕ n+1 (y) : R, or K ( ) (n + 1)th Gibbs = exp X= X i i g D R X i [ ] Wn+1 K (ϕ n+1) F K \X (ϕ n+1 ) X i ξ D R X i X j =, X i D R F K \X (ϕ n+1 ) = X ϕ n+1 ( ) Nov., / 45

27 2 W K n+1 W K n+1 = 1 2 ϕ n+1, G 1 n+1 ϕ n N : ϕ2 n+1 : G n+1, D n+1 : ϕ 2 n+1 : G n+1 K γ n ϕ 2 n+1, E G 1 n+1 E ϕ 2 n+1 K (i) D n+1 D > 0. (ii) g D R X i X gd R X i D n D > 0. (D, R ) ( ) Nov., / 45

28 2 次元シグマ模型と繰り込み群 1/2 球面に拘束される揺動場 zn = QΓn ξn ] [ g exp ((ϕn+1 + zn )2 Nβn )2 かつ ϕ2n+1 Nβn+1 = O(1) N 揺動場 ξn (x) は背景場のブロックスピン ϕn+1 に影響される. ξn は井戸の底に生きてかつ ϕn+1 に鉛直にすすむ. ξn T (S N 1 ). 伊東恵一 (立教大学数理物理学研究センター) 繰りこみ群とミレニアム問題 Nov., / 45

29 Mathematical Meanings of RG D(ϕ n ) = Large and/or non-smooth configuration of ϕ n = D 0 (ϕ n+1 ) D w (ϕ n+1 ) R(z n = Qξ n ) = + D w = ϕ n (x)ϕ n (y) Nβ n > τ 0 N 1/2 exp[(c/10) x y ] x D w, y D w 1/2 : ξ 2 i : D w ϕ n (x)ϕ n (y) Nβ n < τ 0 N 1/2 exp[(c/10) x y ] x D c w, y D c w (D w ) c ϕ n (x)ϕ n (y) = NG n (x, y) ( ) Nov., / 45

30 Mathematical Meanings of RG ϕ n = A n+1 ϕ n+1 + Qz n )A n+1 ϕ n+1 Qz n ( ) Nov., / 45

31 Mathematical Meanings of RG = ϕ n (x) ξ n (x) N 1 (tangent vector ) ( ) Nov., / 45

32 Mathematical Meanings of RG : Z (x, y) = : ϕ n (x)ϕ n (x) : : φ n+1 (x)φ n+1 : = φ n+1 (x)z n (y) + φ n+1 (y)z n (x)+ : z n (x)z n (y) : K = {{ξ n (u) R N ; u K }; Z (x, y) τ 0 N 1/2 exp[(c/10) x y ]} = ( ) ( ) Nov., / 45

33 Mathematical Meanings of RG RG Banach Space H K n K 1 K 2 K n K n = (.1 ϕ n (x)ϕ n (y) Nβ n < τ 0 N 1/2 exp[(c/10) x y ] x, y K.2 ϕ n (x) 2 Nβ n < τ 0 N 1/2.3 ϕ n (x) < τ 0 N 1/2 ( ) Nov., / 45

34 Mathematical Meanings of RG.1 ϕ 4 ϕ n (x) = A n ϕ n+1 + Qξ(x) ϕ n+1 ([x/l]) + Qξ(x) ϕ n (x), x < L/2 L 2 ϕ n+1 ([x/l]) ϕ 2 n(x) L 2 ϕ 2 n+1 (x) x x (: ϕ 2 n : Gn (x)) 2 L 2 (: ϕ 2 n+1 (x) : G n+1 ) 2 x x ϕ 4 n,..2 ( ), ( ) Nov., / 45

35 Main Theorem on RG flow Gibbs ϕ n+1 K / D, : ξ. K W n Wn K (ϕ n, ψ n ) = 1 2 ϕ n, Gn 1 ϕ n + g n 2N : ϕ2 n : Gn, : ϕ 2 n : Gn γ n < ϕ 2 n, E G 1 n E ϕ 2 n >.1 G 1 n = + m 2 n, m 2 n = L 2n m γ n = (Nβ n ) 1..3 g n g = O(1) > 0 ( ).4 E = N (C) = {f ; Cf = 0} ( ) Nov., / 45

36 Main Theorem on RG flow (marginal term) (irrelevant term) (: ϕ 2 n :) 2 (relevant term) g n 3 mn 2 = L 2n m0 2 exp[ 4πβ + 2n log L] O(1), β n = β const.n O(1) γ n = O((β n N) 1 ) g n = O(1) :,. ( ) Nov., / 45

37 3 :.1,.2,.3 ϕ n = φ n+1 + z n, φ n+1 = A n+1 ϕ n+1, z n = Qξ n < ϕ n, G 1 n ϕ n > = < ϕ n+1, G 1 n+1 ϕ n+1 > + < ξ n, Γ 1 n ξ n >, Γ 1 n = Q + Gn 1 Q + ( Λ)Q > O(1) : ϕ 2 n(x) : Gn = : ϕ 2 n+1 (x) : G n+1 +q(x) q(x) = 2ϕ n+1 (x)z n (x)+ : z(x) 2 n : Γn ( ) Nov., / 45

38 : ϕ 2 n(x) : Gn : φ 2 n+1 (x) : G n+1 = q(ξ) = 2φ n+1 (x)z n (x)+ : z(x) 2 n : Γn dµ(ξ) ( q(ξ)dµ = 0, ): P(φ n+1, p) = [ ] i exp (p q(ξ)) x λ x dµ(ξ) dλ x N x [ ] i = exp (p q(ξ)), λ dµ(ξ) dλ x N dµ(ξ) = exp[ ξ, Γ 1 n ξ ] dξ x ( ) Nov., / 45

39 1: p ( ) P(p, φ) = exp[ 1 4N p, 1 M p ] M = Γ 2 n + 2 N (ϕ nϕ n ) Γ n = Γ 2 n + 2β n Γ n + : ϕ n ϕ n : }{{} Γ n /N (Γ n )(x, y) = (QGn 1 Q + )(x, y) exp[ x y ] ((ϕϕ) Γ)(x, y) = (ϕ(x)ϕ(y))γ(x, y) NG(x, y)γ(x, y) spec M = { κ }{{} 0, κ 1,, κ L 2 1} }{{} O(1)>0 O(β n ) ( ) Nov., / 45

40 2:, : ϕ n (x)ϕ n (y) : Γ n (x, y) < N 1/2+ε 1 ( p, 1 M p = 1 p U, E 0 + κ 0 blocks:u Λ n i ( ) 1 p U, E 0 P U κ 0 blocks:u Λ n 1 = (E 0 p U ) 2 κ 0 blocks:u Λ n ) 1 E i p U κ i E 0 = E 1 = ( ) Nov., / 45

41 2 ϕ n (x)ϕ n (y) Nβ n > τ 0 N 1/2 exp[(c/10) x y ] x D w, y D w D w exp[ 1 2 φ n, G 1 0 φ n Dw ]dµ(ξ) < exp[ τ 0 D w ] : g D R, polymer. : D w φφ NG n ( ) Nov., / 45

42 3 exp[ g n 2N : φ2 n :, : φ 2 n : + (...)]dµ(ξ) = exp[ g n 2N : φ2 n+1 : +p, : φ2 n+1 : +p ]P(p, φ) dp P(p) = exp[ 1 4N p, M 1 p ] = exp[ 1 ( ) 1 4N p, E 0 p ] = exp[ 1 (E0 p U ) 2 ] κ 0 4Nκ 0 E 0 p {p} (block spin type.) ( ) Nov., / 45

43 p : p : φ 2 n+1 : 2. g n : g n 2N = g n 2N (: φ 2 n+1 : +p)2 + 1 (E 0 p) 2 2N x x [(E 0 (: φ 2 n+1 : +p))2 + ((1 E 0 )(: φ 2 n+1 : +p))2] + 1 2N x (E 0 p) 2 x E 0 p (1 E 0 )p, steepest descent + ). P(p) = P n (p) n, 4 g n : g n g ( ) Nov., / 45

44 ( ). A.Jaffe and E.Witten, Quantum Yang-Mills Theory, in Millennium Problems, Clay Mathematical Institute. T.Balaban, A low temperature expansion for classical N-vector model I, Commun.Math.Phys., (1995); Variational problems for classical N-vector model, Commun.Math.Phys., 175, 607 (1996) J.Dimock, The Renormalization Group According Balaban, I, arxiv 1108/1335; II, arxiv: J.Fröhlich and T.Spencer, The Kosterlitz-Thouless transitions in Two-Dimensional Abelian Spin Systems and the Coulomb Gas, Commun.Math.Phys., 81, 527 (1981) ( ) Nov., / 45

45 M.Hairer et al., A class of growth models rescaling to KPZ, arxiv: A.Kupiainen, Renormalization of Generalized KPZ equation, arxiv: K.Gawedzki and A.Kupiainen, Massless lattice ϕ 4 4 theory: Rigorous control of a renormalizable asymptotically free model, Commun.Math.Phys. 99, 197 (1985), and references cited therein.,, SGC 81 (2011), K.R.Ito, Renormalization Group Flow of 2D O(N) Spin Model with large N, Absence of Phase Transitions, Paper in Preparation K.R.Ito, Origin of Asymptotic Freedom in Non-Abelian Field Theories, Phys.Rev.Letters, 58 (1987) 439 ; Renormalization Group Flow of 2D Hierarchical Heisenberg Model of Dyson-Wilson Type, Commun. Math.Phys. 137(1991) 45 ( ) Nov., / 45

磁性物理学 - 遷移金属化合物磁性のスピンゆらぎ理論

磁性物理学 - 遷移金属化合物磁性のスピンゆらぎ理論 email: takahash@sci.u-hyogo.ac.jp May 14, 2009 Outline 1. 2. 3. 4. 5. 6. 2 / 262 Today s Lecture: Mode-mode Coupling Theory 100 / 262 Part I Effects of Non-linear Mode-Mode Coupling Effects of Non-linear