? URL: http://maildbs.c.u-tokyo.ac.jp/ fukushima mailto:hukusima@phys.c.u-tokyo.ac.jp DEX-SMI @ 2006 12 17 ( ) What is scaling theory? DEX-SMI 1 / 40
Outline Outline 1 2 3 4 ( ) What is scaling theory? DEX-SMI 2 / 40
r αr = S(r) S(αr) =? r S(r) S(r) = πr 2 r αr α? S(αr) = π(αr) 2 = α 2 S(r) 2 2 = d dr log S(r) r=1 ( ) What is scaling theory? DEX-SMI 4 / 40
S(r) r L R(L) R(L)? L R(L) R(L) = R 0 L φ φ:! R 0 ( ) What is scaling theory? DEX-SMI 5 / 40
m r = {r 1, r 2,, r N } m d 2 dt 2 r i = r i U(r 1, r 2,, r N ) (1) U k U(ar 1, ar 2,, ar N ) = a k U(r 1, r 2,, r N ) (2) r i r i = αr i, t t = βt. (3) β 2 α m d 2 dt 2 r i = 1 α k 1 r i U(r 1, r 2,, r N ) (4) ( ) What is scaling theory? DEX-SMI 6 / 40
U(r 1, r 2,, r N ) = mg(z 1 + z N ) k = 1 = β 2 /α = 1 α = 10 β = 10??? 1M T 10M 10 1M 10M 10! ( ) What is scaling theory? DEX-SMI 7 / 40
? 7:30 7 50!!. ( ) What is scaling theory? DEX-SMI 8 / 40
3/2 r? k = 1, = β 2 = α 3 T ( ) R 3/2 T = R 1000 100 10 1 0.1 0.01 Kepler s third law 1 10 ( ) What is scaling theory? DEX-SMI 9 / 40
??. ( ) What is scaling theory? DEX-SMI 11 / 40
?(2). ( ) What is scaling theory? DEX-SMI 12 / 40
ξ L sim ξ L sim ( ) What is scaling theory? DEX-SMI 13 / 40
e.g. (! : ξ (T c ) =. L sim > ξ (T c )! ( ) What is scaling theory? DEX-SMI 14 / 40
T < T c M T c T T c β ξ T T c ν T c, χ T T c γ T c 1 H/Tc=0.0 H/Tc=0.1 0.8 0.6 m 0.4 0.2 0 0 0.5 1 1.5 2 2.5 3 T/Tc ( ) What is scaling theory? DEX-SMI 15 / 40
( ) ex. H(S) = J ij M 1 S i = N M = i {S i =±1} {S i =±1} S i S j = H( S) ( ) 1 exp( βh(s)) S i N Z i ( ) 1 exp( βh( S)) S i N Z i = 1 N i S i = 0, Si ( ) What is scaling theory? DEX-SMI 16 / 40
:. L T A(L, T ), 1 L A(, T )!. 2 ( )!. 3 L ( L L ) INPUT A(L, T ) for 1, 10, 10 2, = OUTPUT A(, T ) and/or T c, ν,..? e.g. A(L, T ) = A(, T ) + a 1 L b 1? + a 2 L b 2 + ( ) What is scaling theory? DEX-SMI 17 / 40
2 : χ(l, T ) = 1 N ( Si S j S i S j ) = r ij S 0 S r, m (2) (L, T ) = χ(l, T ) N χ M (2) : L = 64 = N = 4096 L = 4, 8, 16, 32, 64 (T,L) 10000 1000 100 10 1 1 1.5 2 2.5 3 3.5 4 4.5 5 T/J 64 M 2 (T,L)/V= 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 1 1.5 2 2.5 3 3.5 4 4.5 5 T/J 10000 1 L= 4 L= 4 8 0.9 8 16 16 1000 32 0.8 32 ( ) 64 What is scaling 0.7 theory? DEX-SMI 64 18 / 40 64
by my M. N. Baber, in Phase transitions and Critical Phenomena, edited by C. Domb and J. L. Lebowitz, vol. 8 (Academic Press), 1983. D.J.Amit, V.Martín-Mayor, Field Theory; The Renormalization Group and Critical Phenomena. 3rd Edition ( ) What is scaling theory? DEX-SMI 20 / 40
ξ (T ) L/ξ (T ) L/ξ (T ) 1: L/ξ (T ) 1: L L/ξ ( ) A(L, T ) L φ A L Ã ξ (T ). ξ (T ) φ A A(L, T ) ( ) A(L, T ) L L φ Ã A ξ (T ) T ( ) What is scaling theory? DEX-SMI 21 / 40
2 ξ (T ) T T c ν T c for T Tc ( A(L, T ) L φ A à L/ T T c T c = T T c T c ( Aν φ L/ ν ) For A = χ(susceptibility), χ (T ) T T c T c T T c T c ν) φa à ( L/ T T c T c γ for T Tc φ A = γ/ν. = 3 T c, ν, γ/ν Ãχ(x) x φ A for large x. ν ) ( ) What is scaling theory? DEX-SMI 22 / 40
Finite-Size Scaling Example for 2D Ising model Raw data: befor scaling (T,L) 10000 1000 100 10 L= 4 8 16 32 64 1 1 1.5 2 2.5 3 3.5 4 4.5 5 T/J disconnected susceptibility : χ(l, T ) = 1 N ij S is j When correlation length reaches size L, this susceptibility gets a constant. χ FSS plot: L 2 η vs (T T c)l 1/ν. (T,L)/L 2 2.5 2 1.5 1 0.5 L= 4 8 16 32 64 0-10 -5 0 5 10 ( T 1)L 1/ Tc choose T c, γ/ν and ν so as to get a universal function. A deviation from the universal function is attributed to the correction to scaling... ( ) What is scaling theory? DEX-SMI 23 / 40
. 1 T = T c : χ(l, T c ) = L γ/ν χ(0) for L. 2 T > T c : ( χ(l, T ) χ L/ T T c L γ/ν T c ν) χ( ) = 0 χ(x) x γ/ν for x 1(L ), ( = χ(l, T ) L γ/ν L/ T T ) γ/ν c χ( ) T c L L( T-Tc /Tc) ( ) What is scaling theory? DEX-SMI 24 / 40
? ξ (T ) T T c ( ) ν ( ( ) ω ) T Tc T Tc ξ (T ) C χ 1 + C 1 + T c T c???? L χ(l, T ) L γ/ν F 0 ( L ((T T c )/T c ) ν )+L ω F 1 ( L ((T T c )/T c ) ν ) + ( ) What is scaling theory? DEX-SMI 25 / 40
Binder parameter 2 Binder parameter: (Phys. Rev. Lett. 47 (1981) 693.) ( U(L, T ) = 1 M4 3 M 2 2 = Ũ L/ T T c T c ν ) φ A = 0 A ξ L /L F (L, T )/T ( ) What is scaling theory? DEX-SMI 26 / 40
Binder parameter 2D Ising model: U(L, T )vs T T c T c L 1/ν 1 0.9 0.8 1 0.9 0.8 g(t,l) 0.7 0.6 0.5 0.4 0.3 L= 4 0.2 L= 8 L=16 0.1 L=32 L=64 0 1.8 1.9 2 2.1 2.2 2.3 2.4 2.5 2.6 T/J g(t,l) 0.7 0.6 0.5 0.4 0.3 L= 4 0.2 L= 8 L=16 0.1 L=32 L=64 0-10 -5 0 5 10 (T/Tc 1)L 1/ ν ( ) What is scaling theory? DEX-SMI 27 / 40
ξ(l, T ) ξ (T )! ξ(l, T )? A = ξ ( ) ( ) ξ(l, T ) L φ ξ L F = ξ φ ξ L φξ ( ) L F ξ (T ) ξ ξ (T ) ( ) ( ) = ξ φ ξ F L = T T c φ ξν L F ξ (T ) ξ (T ) L φ ξ = 1 F ( ) = ( ) ξ(l, T ) L F L = ξ ( ) (T ) L F ξ (T ) L ξ(l, T ) ξ /L ξ(l, T )/L ξ(l, T )! ( ) What is scaling theory? DEX-SMI 28 / 40
2 FSS ( ) A(L, T ) L φ A ξ(l, T ) Ã L T c ν = ( ) χ(l, T ) ξ(l, T ) L γ/ν Ã L INPUT: χ(l, T ), ξ(l, T ) OUTPUT: γ = (T,L)/L 2 10 0 10 1 10 2 10 3 L= 4 8 12 16 24 32 48 64 10 1 10 0 10 1 (T,L)/L ( ) What is scaling theory? DEX-SMI 29 / 40
Campbell-Hukushima-Takayama: Phys.Rev.Lett.97, 117202 (2006). β ξ (1 T c /T ) ν = χ(l, T ) L γ/ν ( ) ξ(l, T ) F χ β L T] 2 (T,L)/[L 10 0 10 1 10 2 L= 4 8 12 16 24 32 48 64 10 3 10 2 10 1 10 0 10 1 10 2 (T,L)/L ( ) What is scaling theory? DEX-SMI 30 / 40
A(L, T ) A(, T )?!?? (?) J-K.Kim(1994), S.Caracciolo et al(1995). A(L, T ) ( ) L A(L, T ) = A(, T )f A ξ(l, T ) ξ(l, T ) ( ) L ξ(l, T ) = ξ (T )f ξ ξ(l, T ) ( ) What is scaling theory? DEX-SMI 31 / 40
(1) by Kim ( ) ξ(l, T ) ξ(l, T ) ξ (T ) = f ξ L Step0: /L 25 20 15 10 5 L= 8 L=12 L=16 L=24 0 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 T/J Step1: ξ(l, T ) vs ξ(l, T )/L. (L,T) 2.05 2 1.95 1.9 1.85 1.8 1.75 1.7 Fixed T T/J=1.80 T/J=1.77 0 0.05 0.1 0.15 0.2 0.25 (L,T)/L ξ(l, T ) vs ξ(l, T )/L L! ( ) What is scaling theory? DEX-SMI 32 / 40
(2) Step2: ξ(l,t ) ξ(,t ) vs ξ(l, T )/L (T) (L,T)/ 1.1 1.05 1 0.95 0.9 0.85 0.8 T/J=1.80 T/J=1.77 0 0.05 0.1 0.15 0.2 0.25 (L,T)/L Step1 y y ( ) T ξ (T ) A(L, T ) OUTPUT:A(, T )?? ( ) What is scaling theory? DEX-SMI 33 / 40
(3): Scaling Plot (T) (L,T)/ 1.1 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 T/J=1.8 1.703 1.609 1.519 1.403 1.374 1.345 1.317 1.289 1.260 1.232 1.204 1.176 1.148 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 (T,L)/L L/ξ 1: L/ξ O(1):! (L,T) 9 8 7 6 5 4 3 2 1 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 1/L T/J=1.80 T/J=1.61 T/J=1.40 1/L ( ) What is scaling theory? DEX-SMI 34 / 40
(4): ξ (T ) 1000 100 L= 8 L=12 L=16 L=24 (T) 10 3 10 2 10 1 Tc=1.106(4) =1.95(6) /L 10 10 0 0.1 1 T Tc 1 0.8 1 1.2 1.4 1.6 1.8 2 2.2 T/J? ξ (T T c ) ν T c 1.1 ν 1.95 ( ) What is scaling theory? DEX-SMI 35 / 40
FSS Carraciolo et al, PRL(1995)? ( ) L ξ(l, T ) = ξ (T )f ξ, ξ(l, T ) ( ξ(2l, T ) = ξ (T )f ξ 2L ) ξ(l, T ) ξ(2l, T ) = f ξ ( f ξ ( L ξ(l,t ) 2L ξ(2l,t ) ξ(2l, T ) ) = f ξ ( L ξ(l, T ) ( ) A(L, T ) L A(2L, T ) = Ã ξ(l, T ) ) ),!! ( ) What is scaling theory? DEX-SMI 36 / 40
ξ(l, T ) 4 χ(l, T ). 4 (2L,T) (L,T 3.5 3 2.5 2 1.5 L= 4 1 L= 8 L=12 0.5 L=16 L=24 L=32 0 0 0.5 1 1.5 2 2.5 3 (L,T)/L (L,T) (2L,T) 3.5 3 2.5 2 L= 4 1.5 L= 8 L=12 1 L=16 L=24 L=32 0.5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 (L,T)/L (L 2L) ξ(l, T ) χ(l, T ) ξ(2l, T ) χ(2l, T ) ξ(4l, T ) χ(4l, T ) ξ(, T ) χ(, T ) OUTPUT:ξ (T ), χ(, T ). ( ) What is scaling theory? DEX-SMI 37 / 40
1 ξ(, T ) Kim s method Carraciolo s method T c γ. χ(t ) vs T /T c 1 (T,L) 1000 100 10 L= 4 8 16 32 64 Carraciolo s method Kim s method 1 0.01 0.1 1 T/Tc 1 ( ) What is scaling theory? DEX-SMI 38 / 40
? ( ) ( ) correction-to-scaling? scaling variables? ( )? ( ) What is scaling theory? DEX-SMI 40 / 40