格子QCD実践入門
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1 -- nakamura at riise.hiroshima-u.ac.jp or nakamura at an-pan.org
2 1. vs. 2. (1) 3. QCD QCD QCD 4. (2) 5. QCD 2
3 QCD 1981 QCD Parisi, Stamatescu, Hasenfratz, etc 2 3
4 (Cut-Off) = +Cut-Off a p max = π a, etc. 4
5 Wilson Wilson, Confinement of quarks, Phys.Rev. D10 (1974) 5
6 Parisi 1982 QCD Creutz, Confinement and the critical dimensionality of space-time, Phys. Rev. Lett. 43 (1979)) Stochastic 6
7 QCD J P t J P 7
8 QCD U: ψ Δ
9 µ=x,y,z,t or 1,2,3,4 x 1 =1, 2,,N x x 2 =1, 2,,N y x 3 =1, 2,,N z x 4 =1, 2,,N t x µ
10 (x) (x) (x) (y) (x 6= y) (x)e i R y x A µdx µ (y) 10
11 e i R y x A µdx µ y x e iaa a e iaa µ U µ (x)u (x 0 ) U (y) 11
12 UU U Pe i R y x A µdx µ 12
13
14 みな という形 ウイルソンの格子ゲージ理論 ただし向きが逆の時は x
15 15
16 ( 格子上の ) ゲージ変換 x y!!(x) U j (x)!(y) ただし!(x) =e ie (x)
17 U µ (x) (x) (x) U µ (x) (x +ˆµ) (x) (x) (x) (x) (x) (x) (x) (x)uµ (x) (x +ˆµ)
18 Tr U ij U jk U kl U li
19 U(1)
20 SU(N)
21
22 x 2 x 1
23 N: N N=1000 n=10 QCD
24
25 Importance Sampling x t
26 Importance Sampling (2)
27 Metropolis Importance Sampling + Random Sampling N.Metropolis et al. J. Chem. Phys. 21, 1087 (1953) Web p1087_s1?bypasssso=1
28 Start DO S(x) x End
29
30 Metropolis Algorithm s : s s exp( E s /kt ) Nicholas Metropolis Νικόλαος Μητρόπουλος P rs P rs = P sr r s ( ) Ergodic ( ) r s E r >E s r s s r rp rs sp sr exp( (E r E s )/kt ) 30
31 s-->r, r-->s rp rs = sp sr exp( (E r E s )/kt ) P rs = P sr r exp( E r /kt ) = s exp( E s /kt ) s r constant s = C exp( E s /kt ) E r <E s 31
32 M. Creutz and B. Freedman, Ann. Phys , (1981) t tf x n-1 ti x x 2 x 1
33
34
35 τ τ f x n-1 τ i x 2 x 1 V(x) x x
36
37 V(x)
38 MAIN DO isweep=1, Nthermal CALL update DO isweep=1, Nsweep CALL update CALL measurement
39 update DO i = 1, N x_old = x(i) x_new = x_old + α*(r1-0.5) ds=s(new)-s(old) Metropolis check: x(i) = x_old or x_new
40 x N+1 =x(1), x( N) x N+1 =-x(1), x( - N) DO i = 1, N ia = i + 1 ib = i - 1 IF( i==n ) ia = 1 IF( i==1 ) ib = N xa = x(ia) xb = x(ib)... REAL, DIMENSION(0:N+1) :: x x(0) = x(n) x(n+1) = x(1) DO i = 1, N xa = x(i+1) xb = x(i -1)...
41 INTEGER, DIMENSION(N,2) :: inn SUBROUTINE MakeTable DO i = 1, N xa = x(inn(i,1)) xb = x(inn(i,2))... DO i = 1, N ia = i + 1; ib = i 1 IF( i==n ) ia = 1 IF( i==1 ) ib = N inn(i,1) = ia inn(i,2) = ib ENDDO RETURN END
42 境界 : 固定 自由 周期 反周期 +: 周期 ー : 反周期 あるいは
43 ( 反 ) 周期境界条件 ( 周期 ) ( 反周期 )
44 Overlap Problem Importance Sampling Importance Sampling O = Of(x)dx f(x)dx 44
45 QCD ( ) a:
46
47 QCD K.G.Wilson Phys. Rev. D10, 2445 (1974) Erice Lecture Note 1977 NxNyNzNt
48 スピン型相互作用とゲージ相互作用 スピン型 というタイプの相互作用から現れる
49 ゲージ型 ループ型のため 斜め横の点に属するものとも相互作用 並列化の時に注意を要する
50 S F = i,j i (i, j) j (i, j) ab (i, j) : hopping parameter
51
52 U(1)
53
54 (naïve classical limit) QCD OK O(a) Improved action Iwasaki Syzmanzik (DBW2
55 Wilson Loop Polyakov Line
56 external source or
57 T L
58 Polyakov Line Polyakov line: Confinement
59 Z3 SU(3) 1,e i 2 3,e i 4 3 U t (i),u t (j),,u t (k), Z3
60
61 M.Creutz, Phys.Rev.D21, 2308 (1980) SU(2)
62
63
64
65 KS (Kogut-Susskind) Wilson fermions r: Wilson c=2ma
66 (1) (2) A µ e i e+i B e +i 5 e+i 5 (1) B
67 (1981)
68 Ginsparg-Wilson (1982) Neuberger (1998)
69 = Gauss N^3 N (Sparse
70 X = b b = i X = =( 1 1 ) b i 70
71 Conjugate Gradient Method, CG Ax = b (x, Ax) 0 A: for t AAx = t Ab f(x) = 1 2 (x, Ax) (b, x) f(x) f(x) =Ax b = 0 x
72 CG DO i p (0) = r (0) = b Ar (0) r (i) = b Ax (i) (i) = (p(i), r (i) ) (p (i),ar (i) ) x (i+1) = x (i) + r (i+1) = r (i) p (i+1) = r (i+1) + (i) p (i) (i) Ap (i) (i) = (r(i+1),ap (i) ) (p (i),ap (i) ) (i) p (i) Residue, p (1), p (2), p (3), r (1), r (2), r (3), N (Matrix) (Vector)
73 0 Berezin (1966) Matthews-Salam
74 For Exercise " ψ #" A A # ψaψ = $ %$ % ψ ψ ψ A A & 2'& 21 22' ψaψ ( ) 1 2 Show that dψ dψ dψ dψ e = det A ψaψ e = 1 + ( ψ A ψ + ψ A ψ + ψ A ψ + ψ A ψ ) ( 2 ψ1a11ψ1 ψ1a12ψ ).. 2 ψ2a21ψ2 ψ2a22ψ Only these terms contribute 74
75 1 Z DUDūDuD ddd e S G ū u d d (x) (y) = 1 Z DUe S G det (u) det (d)
76 1 Z DUe S G det (u) det (d) x y G (u) ( (u) ) 1 ( G (d) (d) ) 1
77 sigma 1 Z DUDūDuD ddde S G ū u d d
78 1 Z DUe S G det (u) det (d) x y x y
79 79
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0. A A = 4 IC () det A () A () x + y + z = x y z X Y Z = A x y z ( 5) ( s5590) 0. a + b + c b c () a a + b + c c a b a + b + c 0 a b c () a 0 c b b c 0 a c b a 0 0. A A = 7 5 4 5 0 ( 5) ( s5590) () A ()
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