LGT U x, ˆ µ = U 11 L L M O M M L U Nc N c dµ(u) = e β UUU + U + x, ˆ µ det / D (U) + m x, ˆ µ ( ) N F du x, ˆ µ RMT H = H 11 L L M O M M L H NN dµ(h) = e tr H 2 dh
LGT RMT
Akemann-Damgaard-Magnea-SMN 97/98 SMN 98/99, Garcia-SMN-Verbaarschot 02 Nagao-SMN 01 Damgaard-SMN 98-01, Nagao-SMN 00/01 Dunne-SMN 03
s 1950s
H s P Wigner (s) = π 2 s e- π 4 s 2
3 3
n n 1 P Poisson (s) = e -s
V(x,y) = A x 4 + B y 4 + C x 2 y 2
2
ζ(1/2 + i x) x 10 20 th ~ (10 20 + 10 8 )-th zeroes 2 P(s)
H Prob(E,E ) ~ E -E β β=1,2,4 H
T = K C c.c. unitary T 2 = CC * =±1 symm. C = U T U antisymm. C = U T JU [H, T] = 0 H : R symm H : H selfdual L S [H, T] 0 H : C hermitian S B
SU(2)
# # L # # # H = # # # # # # # # M # O H 11 H 12 H 13 H 14 H 15 H 16 L H 21 H 22 H 23 H 24 H 25 H 26 L H 31 H 32 H 33 H 34 H 35 H 36 L H = H 41 H 42 H 43 H 44 H 45 H 46 L H 51 H 52 H 53 H 54 H 55 H 56 L H 61 H 62 H 63 H 64 H 65 H 66 L M M M M M M O
H = H + = (H ij ) R, C, H dµ(h) = exp(-tr H 2 ) Πd β H ij β = 1, 2, 4 dµ(h) = dµ(uhu + ) dµ({e}) = Π i de i exp(-tr E i2 ) Π i>j E i - E j β
N=2 N=
N=11, 21, 51 N= ( N=50 ) x 100
2 N=2 3λ 1 λ 2 3
N C =1, N F =0, 1-plaquette
2 β=0 β=1 β=2 β=4 exp(-s) ~ s β ~ exp(-c β s 2 )
s k E β (k ; s) GOE Poisson GUE
N P (s), ρ(λ), N= P (s), ρ(λ), exp(-tr H 2 ) exp(-tr V(H)) exp(-tr (H+A) 2 ) Anderson Gauss Anderson P (s)
Vector subgroup Sp(2n)
Riemann 1 U = V diag(e iθ 1,...,eiθ N ) V + Cartan Zirnbauer 96
λ µ µ U 1 nonchiral U/UxU U/O 1/2 nonchiral Sp/SpxSp U/Sp 2 nonchiral O/OxO U/UxU 1 Z+1/2 1/2 U O/OxO 1/2 Z/2 0 U/Sp Sp/SpxSp 2 2Z+3/2 3/2 U/O O 1 0, 1 0 O/U Sp 1 1 1 Sp/U Sp/U 1/2 1/2 1/2 Sp O/U 2 1/2, 5/2 1/2 O Zirnbauer 96
N n N=n=1 any N, n U(1) dual C = U(2)/U(1)xU(1) Gauge U(N) dual U(2n)/U(n)xU(n) Meson Witten 79
det(z U) = du = du det(z U) exp( Ψ f f i (zδ ij U ij )Ψ ) j N N L LS [Z]
-
V x V(x) t t W x t W
t >> V 0 -V 1 t << V 0 -V 1 ψ x ψ x t > W Prob( ψ 0 ~ ψ n ) = 1 t < W Prob( ψ 0 ~ ψ n ) = (t/w) n 0
g(l) L L+dL g(l+dl) = F(g(L), (L+dL)/L) dlog g/dlog L = β(g) D>2 - β(g) g*. 0 g D>2 D=2 D<2 g(l) ~ exp(-l/ξ) β(g) ~ log g L g(l) ~ L D-1 /L β(g) ~ D-2
δx 2 = D δτ >> L 2 /D ( ) 1 = h/ << E c hd / L 2
E ψ (x) 2 ξ >> L ξ ~ L ξ << L x
Σ ψ 2p ~ V -d p(p-1) Z(E-E ) = Σ ψ Ε 2 ψ Ε 2 ~ E E -(1-d 2) D=3 β=1 β=2 ( )
Σ ψ Ε 2 ψ Ε 2 ~ E E -(1-d 2) δn 2 ~ χ L, χ = (1-d 2 )/2 p(s) ~ exp(-κ s), κ = 1/(1-d 2 ) ( ) ( )
D=3 (β=1 ) ~ s 1 ~ exp(-s/2χ) ~ log L ~ χ L
L(Q) = 1/(V ) [D tr ( Q) 2 + δe tr ΛQ], Q(x) : NG E c 0 E c >> E c ~
( ) H = (H ij ) = 1/(1+a 2 (i-j) 2 ) H UHU + d p ~ 1 - (a/2π 2 )p (g*>>1) d p ~ 1 - (1/g*)p 1 a
β=1 β=2 ~ s β ~ e -s/2χ β=4
D=3 Anderson
D=3 Anderson D=2 Anderson : Evangelou β=1 : β=4 ~ log L ~ χ L
[ ] [ ] [ ]
[ ] g*
D=4 vs Garcia-Verbaarschot β=2
D=3
g* L β=2 Σ 2 (L) ~ χ L, χ
g* g*>>1
: << Λ QCD QCD [π] : ε
initiated by Jac Verbaarschot Stony Brook + Akemann G, Altland A, Berbenni-Bitsch ME, Berg BA, Bietenholz W, Bittner E, Dalmazi D, Damgaard PH, Edwards RG, Farchioni F, de Forcrand F, Fyodorov YV, Garcia-Garcia AM, Giusti L, Gockeler M, Guhr T, Halasz MA, Hehl H, Heller UM, Hilmoine C, Hip I, Iida S, Jackson AD, Janik RA, Jansen K, Jurkiewicz J, Kaiser N, Kalkreuter T, Kanzieper E, Kiskis J, Klein B, Krasniz A, Lang CB, Luscher M, Lombardo M-P, Ma J-Z, Madsen T, Magnea U, Markum H, Meyer S, Nagao T, Narayanan R, Niclasen R, Nishigaki SM, Nowak MA, Osborn JC, Papp G, Pullirsch R, Rakow PEL, Rabitsch K, Rummukainen R, Schafer A, Schnabel M, Schwenk A, Seif B, Sener MK, Schlittgen B, Simons BD, Shcheredin S, Shrock RE, Shuryak EV, Smilga AV, Stephanov MA, Splittorff K, Toublan D, Takahashi K, Vanderheyden B, Weidenmuller HA, Weitz P, Wettig T, Wilke T, Wittig H, Wohlgenannt M, Zahed I, Zirnbauer MR, + many others RMT LGT
ψ D / ψ = ψ + L D µ σ µ ψ L +ψ + R D µ σ µ ψ R N F ψ L U L ψ L U L SU(N F ) L ψ R U R ψ R U R SU(N F ) R (N C, N F ) ψ ψ = ψ R + ψ L + ψ L + ψ R 0 ψ L Uψ L, ψ R Uψ R U SU(N F ) V m q 0
Banks Casher 80 d 4 x ψ (x)ψ(x) = tr 1 D / + m = 1 = iλ n + m 2m λ 2 n + m 2 λ n >0 V a = 1 2m dλ ρ(λ) 0 λ 2 + m 2 a 1 2m dλ ρ (λ) 0 λ 2 + m 2 π ρ (0) m 0
Σ ψ ψ = π ρ (0) V = π 0 = O(V 1 )=O(L d ) V = 0 = O(L 1 ) free
V π ψ ψ SU(3), N F =0, staggered V=4 4 Gockeler et al 99
SU(2), N F =0, staggered V=10 4 Berbenni et al 97 ρ s (ζ) = 1 ρ ζ
SU(3), N F =0, staggered V=4 4 Damgaard et al 98 ρ(0) β SU, SO, Sp
probe fermionic & bosonic quarks Z({m f },m m ) = [da] e S YM [ A] f det( / D + m f ) det( D / + m) det( D / + m ) m log Z({m },m m ) f m= m = tr 1 m + / D {m f } m iλ, Im ρ(λ) Z graded
1 Λ QCD << L π Z U =U R U L + : SU(N F ) L SU(N F ) R SU(N F ) V ψ + L Mψ R + c.c. + U u R Uu L + M u L Mu R L chpt = f π 2 tr µ U µ U + Σ Re tr MU +L Weinberg 67
Σ= m log Z m = ψ ψ quench L chpt = f π 2 tr µ U µ U + Σ Re tr MU +L f π 2 L 2 Σ m E C f 2 π >> m,λ Z 2 chpt ΣL ε Leutwyler Smilga 92
1 L << m π Σ level spacing Thouless E hadron mass
/ D Z = dhdψ dψ exp tr H + H + ψ f f R ψ L f = dh e tr H + H f det m f ih + ih m f ( ) m f ih + ih m f ψ f L f ψ R H ψ L f, ψ L f ψ R f, ψ R f N x (N+v) N+v N N F
D / C SU R Sp H SO N F ν N L LS = Re tr MU
Z = dhdψ dψ exp H * f ij H ij + ψ,i f,i R ψ L f ( ) m f ih ji ih ij m f * ψ L f,i ψ R f,i f = dψ dψ exp (ψ,i R ψ g,i L )(ψ g,j f L ψ,j f R )+ m f (ψ,i f R ψ,i f L )+ (ψ,i f ( L ψ,i R )) f = dqdψ dψ exp Q * f fg Q fg + (iq fg + M fg )(ψ,i L ψ g,i R )+ (iq * f fg + M fg )(ψ,i f R ψ,i L ) = dq e -tr Q+Q det N (Q im )det N (Q + im ) { } Q: N F_ x _ N F N N 2 F Z = du e Re tr UM resc U: N F_ x _ N F
dµ(h ) = dh e tr H +H Π f det( H + 2 H + m ) f EV( / D ) =±i EV(H + H ) = { ±i λ 1,...,±i λ N,0,...,0} dµ(λ) =Π{ dλ i e λ 2 i Π( λ 2 2 i + m ) f λ } β(ν+1) 1 Π i λ 2 2 i λ β j i f i> j =Π{ dz i e z i Π( 2 z i + m ) f z } β (ν+1)/2 1 Π i i f i> j z i z j β z i 0 (βν/2) ν
1~4 N F =0, C hermitian Damgaard SN 01
quark mass N F = 3 C hermitian Damgaard SN 98
N F = 1 R symmetric N F = 2 H selfdual Nagao SN 00
2 SU(2), N F =0, staggered V=8 4 Berbenni et al 97
SU(2), N F =4, staggered V=8 4 Berbenni et al 98, Akemann Kanzieper 00 µ m q ρ(0)/π
SU(2), N F =4, staggered, V=8 4 Berbenni et al 98
SU(2) SU(3) SU(3) adj ν=0 ν=1 N F =0, overlap V=4 4 Edwards et al 99
SU(3), N F =0, overlap V=10 4 Bietenholz et al 03 ψ ψ = (256 MeV) 3 (L =1.23 fm)
large physical size 1.23 fm 0.98 fm β=5.85 E c ~ 1.2 fm small physical size
: Damgaard-SN prediction 00 from chrmt SU(3), N F =0, overlap V=20 4 Giusti Luscher et al 03 (L =1.49 fm)
N F =0, overlap V=4 4 Edwards et al 99
confine β =5.2 deconfine β =5.4 SU(3), N F =3, staggered V=6 3 x 4, ma=.05 Bittner et al 00 free β= confine β =0.9 Coulomb β =1.1 U(1), N F =0, staggered V=8 3 x 6 Bittner et al 00
µ ψ + ψ = µ ψ γ 0 ψ D / D / + µγ 0 det( / D + µγ 0 + m)
/ D + µγ 0 SU(3), quenched β=5.2 (confine) V=6 3 x 4 Berg et al 00
Hanano Nelson 96 ih ih + ih + µ ih + + µ Z = dh e tr H + H f m det f ih + + µ ih + µ m f Stephanov 96 Splittorff Verbaarschot 03
µ=0.53 Halasz et al 97
SU(3), N F =0, staggered V=8 4 β=5.0 (confinement) Akemann Wettig 03
µ µ T
q i q j 0 ee 0
µ ψ + ψ = µ ψ γ 0 ψ D / D / + µγ 0 Boltzmann det( D / + µγ 0 + m) µ
Sp(N C ), SO(N C ) [ D / + µγ 0 + m,t ] = 0 ( / D + µγ 0 + m) ab det( / D + µγ 0 + m) R R H gauge spinor c.c. T = JCγ 5K, T 2 =±1 Cγ 5 K Jσ 2 ψ L * σ 2 ψ L * ψ R SU(N F ) L x SU(N F ) R SU(2N F )
qq SU(N C 3) det( / D + µγ 0 + m) R : µ Sp(N C ), SO(N C ) det( D / + µγ 0 + m)det( D / + µγ 0 + m) * det( D / + µγ 0 + m) R qq : µ Kogut Stephanov Toublan 99~01
µ f Z = dh e tr H + H f m det f ih + µ f ih + + µ f m f L LS
G = SU, N F = 2 Klein Toublan Verbaarschot 03 1st order 2nd order
2N F {ψ f, χ f } L mass = m ( f ψ + f ψ f χ + f χ ) f f det( D / 2 + m 2 f ) 0 P x Z 2 (ψ, χ )- m f m 0 f ( ψ + f ψ f χ + f χ ) f SU(2N F ) SU(N F ) x SU(N F )
Sp Jσ 2 ψ T ψ gauge spinorflavor ψ χ L kin = 1 2 ΨT (Jσ 2 D / I)Ψ, Ψ= Jσ 2 ψ T Jσ 2 χ T SU(2N F ) Sp(4N F ) SO J 1, Sp(4N F ) O(4N F )
Sp 1 L mass = 1 2 ΨT (Jσ 2 M ˆ )Ψ, M ˆ = 1 1 Sp(4N F ) Sp(2N F ) x Sp(2N F ) 1 SO J 1, Sp O, ˆ M 1 1 1 1
Sp(N C )- Dunne-SMN 03 L = 1 2 ΨT (Jσ 2 D / I)Ψ+ m 2 ΨT (Jσ 2 M ˆ )Ψ µ 2 ΨT (ijσ 1 C ˆ )Ψ 1 1 C ˆ = 1 1 Sp(N C ), O(N C )-, O (N C )-
1 Λ QCD << L π Sp(4N Σ : F ) Sp(2N F ) Sp(2N F ) m 0, µ 0
L LS
L st =L LS cosα = min 1, 4µ2 2 m π
L st [Σ]
Ω LG ({σ}; m,µ,t ) = L LS ({σ };m,µ) condensate modes + Tr log ({σ};m,µ) t [0, 1/T ] Splittorff Toublan Verbaarschot 02
QCD 3 G = Sp, N F = 2 Dunne SMN 03 tricr 1st 2nd QCD 3 QCD 4 µ m π 2 ~ T logt ~ T 3/2
L eff [Σ;µ,m] F π mass ~ exp(- F π /T) may be generated
chiral symmetry kinematics Dirac ψ ψ