09kyoto

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1 F [ ] F [ ]

2 Nambu( 60), Goldstone(61), Nambu Jona-Lasinio( 61), Goldstone, Salam, Weinberg( 62). N NG = N BS NG :

3 CC by-sa Aney CC by Zouavman Le Zouave CC by-sa Roger McLassus

4 U(1) CC by-sa Aney -Goldstone U(1) 1- CC by Zouavman Le Zouave CC by-sa Roger McLassus

5 U(1) CC by-sa Aney U(1) 1- CC by Zouavman Le Zouave CC by-sa Roger McLassus

6 U(1) CC by-sa Aney U(1) 1- CC by Zouavman Le Zouave CC by-sa Roger McLassus

7 Topological Insulator Electron(up spin) Electron(down spin)

8 ) Goldberger-Treiman relation g NN =2m N g A /f g NN A µ 5 vertex

9 Bloch T 3/2 T 3,... Gadolinium 160r , o (T/T c )3/2 :! k 2 :! k

10 : h[q a, i(x)]i tr [Q a, i(x)] 6= 0 : = ih : = exp( (H µn)) tr exp( (H µn)) well-defined [iq a, ] =0 h[iq a, i(x)]i =tr [iq a, i(x)] =tr[,iq a ] i(x) =0 ill-defined.

11

12

13

14

15 NG N BS 6= N NG! 6= k SO(3)! SO(2) N BS =dim(g/h) =2 N NG =1 :! / k 2 K CFL NG Miransky, Shovkovy ( 02) Schafer, Son, Stephanov, Toublan, and Verbaarschot ( 01) SU(2) I U(1) Y! U(1) em N BS =3, N NG =2 :! / k! / k 2

16 G H N BS =dim(g/h).

17 a N BS

18

19 Nielsen - Chadha( 76) N type-i +2N type-ii N BS Type-I:! / k 2n+1 Type-II:! / k 2n Schafer, Son, Stephanov, Toublan, and Verbaarschot ( 01) h[iq N NG = N a,q b ]i =0 BS Nambu ( 04) h[iq a,q b ]i6=0 (Q a,q b ) canonical conjugate Watanabe - Brauner ( 11) N BS N NG apple 1 2 rankh[iq a,q b ]i

20 Watanabe, Murayama ( 12), YH ( 12) cf. Takahashi, Nitta ( 14), Beekman ( 14) Type-A Type-B N A = N BS rankh[iq a,q b ]i N B = 1 2 rankh[iq a,q b ]i Ex. ) Ex. ) N NG = N BS 1 2 hi[q a,q b ]i

21 Type-A, Type-B z x, y 2

22 Type-A, Type-B

23 Type-A, Type-B 1 {L x,l y } P = L z 6=0

24 Type-A Type-B! p g! g

25 Type-A Type-B! p g p! g k 2 k 2

26 Watanabe, Murayama ( 12) L = 1 2 ab a b + ḡab 2 a b g ab i i b ab / ih[q a,j 0 b (x)]i Watanabe, Murayama ( 12)

27 Type-B NG 1 N BS N type-i N type-a type-ii type-b b ]in BS N 2 rankh[q a,q b ]i N type-a +2N type-b NG SO(3) SO(2) K CFL NG SU(2)xSU(1)Y U(1)em Spinor BEC SO(3)xU(1) U(1) CP U(1)xR 3 R 2 N type-a +2N type-b = N BS N BS N NG = 1 2 rankh[q a,q b ]i

28 Hayata, YH ( 14) Type-A: Type-B:! = ak ibk 2! = a 0 k 2 ib 0 k 4 (or t j 0 = k 2 kj 0 +

29 Hayata, YH ( 14) Type-A: Type-B:! = ak ibk 2! = a 0 k 2 ib 0 k 4 (or t j 0 = k 2 kj 0 + Type-B: k / k 2

30

31 Gaiotto et al. ( 15) i.e., 0- : 1-2- p- p-

32 CP1 Type-B Kelvon [P x,p y ] / N x trans. y trans. 1-form symmetry Type-B Ripplon-Magnon [P z,q] / N z trans. U(1) 2-form symmetry

33 NG Gaiotto et al. ( 15) Wilson ( t Hooft) h W =exp i h H =exp i I I A µ dx µi à µ dx µi : Z Q e = ds E Z Q m = ds B dq e =0 dt dq m =0 dt

34 NG Gaiotto et al. ( 15) Wilson ( t Hooft) h W =exp i h H =exp i I I A µ dx µi à µ dx µi : Z Q e = ds E Z Q m = ds B dq e =0 dt dq m =0 dt

35 NG h i { : : =NG 6 : E i,b i [E i (x),b j (y)] = k (3) (x y) 2 r E =0

36 Yamamoto ( 15) x, y z S = 1 2 Z d 4 x E 2 B 2 + C Z d 4 x E k k 2 z = const Z h[q e (M xz ),Q e (M yz )]i dz@ z -B

37 開放系 CC BY-SA 2.0

38 B. Szabo, et al., Phys. Rev. E 74, (2006) Model of active matter: Vicseck model, Active hydrodynamics,. T. Vicsek, et al., PRL (1995). J. Toner, and Y. Tu, PRL (1995).

39 ) NG J. Toner, and Y. Tu, PRE (1998) + r ( v) t v +(v r)v = v v 2 v rp + D L r(r v)+d l (v r) 2 v + f : : v 2 = / v 2 0 O(3)! O(2) : v =(v 0 + v x, v y, v z )! = ck! = i k 2 NG

40

41 ) d x(t) =u(t) dt d dt u(t) = u(t)+ (t) h i (t) j (t 0 )i =2 ij T (t t 0 ) L = x u d dt hl(t)i = hx u(t)i 6=0

42 d dt u(t) = u(t)+ (t) (or Fokker-Planck ) Z Martin-Siggia-Rose formalism Z = D Due is MSR[,u] is MSR = Z h d dt i i dt u i + u i 2 i 2 i O(3) with i! R ij j u i! R ij u j R ij R kj = jk

43 1) SU(2)xU(1) Type-A V ( ) Langevin (@ r 2 ) a + a Minami, YH ( 15) (@ 2 -A NG 0 0 r 2 ) a =0! 2 i! + k 2 =0! = i 2 ± i p 2 4k2 i k 2, i + i k 2 2

44 Minami, YH ( 15) 2) SU(2)xU(1) Type-B with V ( r 2 2µ@ 0 1 2µ@ r 2 2 =0,! = k 2 4µ (±2µ i ) 2

45 MSR : 2 Type-A Type-B! = ik 2! = ak 2 ik 2 0

46 cf. for review, Sieberer, Buchhold, Diehl, Schwinger-Keldysh 1 Z = Z D 1 D 2 exp h 2 i is[ 1] is[ 2]+iS[ 1, 2] complex Q 1,Q 2 :Symmetry generators: S[ 1],S[ 2]. S 12 [ 1, 2] Q A = Q 1 Q 2 2 Q R = Q 1 + Q 2 2

47 NG 2 Type-A Type-B N A = N BS rankh[iq a,q b ]i N B = 1 2 rankh[iq a,q b ]i! = ak ibk 2! = a 0 k 2 ib 0 k 4

48 k k 2 QA, QR OK k 2 QA Minami, YH ( 15) k 2 k 4 QA, QR OK k 2 k 2 QA

49 Yu, Yang Phys. Rev. Lett (2008), (2010) Shi, Yu, Sun, Phys. Rev. A 81, (2010) Lai and K. Yang Phys. Rev. A 91, (2015) Blaizot, Hidaka, Satow, Phys. Rev. A 92, (2015) Sannomiya, Katsura, Nakayama Gusynin, Miransky, Shovkovy, Phys. Lett. 581B, 82 (2004), Mod. Phys. Lett. A 19, 1341 (2004) Watanabe and Murayama Phys. Rev. D90, (2014) Hama, Hatsuda, Uchino, Phys. Rev. D 83, (2011)

50 = Z d d x hja(x)j 0 a(0)i 0 hja(0)i 0 2 > 0 h[j 0 a(x),j 0 b (y)]i = c ab (r) (x y)

51 Ex1) Kharzeev, Yee µ j µ A = CEi B i [j 0 V (x),j 0 A(y)] = CB k (x y)

52 Ex1) Kharzeev, Yee µ j µ A = CEi B i [j 0 V (x),j 0 A(y)] = CB k (x y) B i 6=0! = v i k i v i = CBi p V A cf. Tomonaga-Luttinger liquid

53 Ex2) h[it 0i (x),t 00 (y)]i = h@ i (x y) h = ht 00 + T ii i

54 Ex2) h[it 0i (x),t 00 (y)]i = h@ i (x y) h = ht 00 + T ii i! = vk v 2 = h 2 T 00 T 00

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