Critical Behavior II: Renormalization Group Theory

Critical Behavior II: Renormalization Group Theor H. W. Diehl Fachbereich Phsik, Universität Duisburg-Essen, Campus Essen 1

What the Theor should Accomplish Theor should ield & explain: scaling laws # of independent critical exponents scaling laws universalit, two-scale-factor universalit determinants for universalit classes clarif to which universalit class given microscopic sstem belongs numericall accurate, experimentall testable predictions crossover phenomena corrections to asmptotic behavior 2

RG Strateg increase minimal length a a = ba such that ξ = ξ additional interaction constants! a a ' = ba ξ ˆ ξ ξ / a ξ ( ξ ) ξ a = a b ˆ ξ = ˆ ξ b large ξˆ: pert. theor fails small ˆ ξ : pert. theor works 3

RG Strateg increase minimal length a a = ba such that ξ = ξ K ( K, h) K ( K, h ) = ij = ij additional interaction constants! a a ' = ba ξ ˆ ξ ξ / a ξ ( ξ ) ξ a = a b ˆ ξ = ˆ ξ b large ξˆ: pert. theor fails small ˆ ξ : pert. theor works 3

Recursion Relations [ ] ( ) K K = R b K 1 [ K ] [ K ] = b [ K ] ˆ ξ ˆ ξ ˆ ξ ˆ ξ important propert: fixed point: R R = R ( b ) ( b ') ( bb ') ( ) K : K = R b K ˆ ˆ * 1 ˆ * ξ ξ K = b ξ K ˆ, T = or ξ K =, critical fixed point 4

RG Flow: 2D Ising-Model K = J / k T ; K = J / k T x x B B K 1+ K J = 2J x T = J = J x / 2 T = K x 1+ K x 5

RG Flow: 2D Ising-Model K = J / k T ; K = J / k T x x B B K 1+ K J = 2 J x T = J = J x / 2 T = K x 1+ K x 5

RG Flow: 2D Ising-Model K = J / k T ; K = J / k T x x B B K 1+ K J = 2 J x T = g τ = g i = J = J x / 2 T = K x 1+ K x 5

Schematic RG Flows in a high dimensional space stable manifold unstable direction all points on this stable basin of attraction flow to the fixed point 6

Linearization [ ] ( ) K K = R b K * K = K +δ K δ δ * ( b) * K + K = R K + K ( δ ) = R K + L δ K + O K ( b) * 2 not in general smmetric δ K = L δ K L ( ) R b j K k * K UɶL U = U δ K ( ) ( ) 1 λρ δ ρρ ; uρ 7

Linearization [ ] ( ) K K = R b K * K = K +δ K * ( b) * K + δ K = R K + δ K ( δ ) = R K + L δ K + O K ( b) * 2 not in general smmetric δ K = L δ K L ( ) R b j K k * K UɶL U = U δ K ( ) ( ) 1 λρ δ ρρ ; uρ linear scaling field ( R b) : u u = λ u ρ ρ ρ ρ RG eigenvalue 7

RG Eigenexponents & Nonlinear Scaling Fields important propert: R R = R ( b ) ( b ) ( bb ) ( b) ( b) R R p times ( λ ) : u u = u ρ ρ ρ ρ p RG eigenexponents ρ λ b ρ : = ρ 8

RG Eigenexponents & Nonlinear Scaling Fields important propert: R R = R ( b ) ( b ) ( bb ) ( b) ( b) R R p times ( λ ) : u u = u ρ ρ ρ ρ p RG eigenexponents ρ λ b ρ : = ρ (+) R b ρ ρ ρ ( ) ρ : u u = b u ρ ρ ρ > : u ± : relevant ρ < : u : irrelevant ρ = : u = marginal ρ nonlinear scaling fields (Wegner): satisf (+) even awa from fixed pt. g = u + C u u + ( ρ ) ρ ρ ρ ρ ρ ρ appropriate curvilinear coordinates 8

reduced free energ densit: Consequences reg sing ( K ) = ( K ) + (,, ) f f f g g 1 2 K = K + δ K d sing ( ) ( ) 1 2 1, g2, = b f g1, g2, sing f g b b ρ choose b such that b g 1, g, 1 = ± 1 > < d / ( ) ( ) sing ϕ ϕ 1, 2,, i = 1 ± 1, 2 1,, i 1 sing 1 2 f g g g g f g g g g i ϕ = i i : 1 crossover exponent if ϕ i < 9

reduced free energ densit: Consequences reg sing ( K ) = ( K ) + (,, ) f f f g g 1 2 K = K + δ K d sing ( ) ( ) 1 2 1, g2, = b f g1, g2, sing f g b b ρ choose b such that b g 1, g, 1 = ± 1 > < d / sing ( ) ( ϕ ϕ ) 1, 2,, i = 1 ± 1, 2 1,, i 1 sing 1 2 f g g g g f g g g g i ϕ = i i : 1 crossover exponent if ϕ i < d / ( ϕ ) sing τ f ( g, g, ; g, ) g Y g g τ h i τ ± h τ h g gτ τ + c δµ + ; g g δµ + c τ + τ h 1,1 2 h 1, sing ( ) = ( 1, ; ) Y g f g g ± h ± h irrelevant = ma be zero or!! dangerous irrelevant variables 9

Scaling Operators g g + δ g ρ ρ ρ d H H d x δ g ρ ( x) O ρ ( x) + ( x) Kadanoff, Patashinski & Pokrovskii O ρ ρ + ( x b) = b O ( x) = d ρ ρ ρ φ( x ) φ( x ) G( x ) = b G( x / b) 2( d h ) 1 2 T 12 12 c 2( d h ) ( d 2 + η ) 12 12 = 12 G( x ) x x 1

1D Ising Model H N 1 N E = = K s s h s + C k T B j j + 1 j j = 1 j = 1 K K K K K K h h h h h h h exact solution periodic bc: ( ) 2 2 Z = Tr e = e e { si =± 1} s T s = TrT ( ) H j 1 j j 1 j j j+ 1 j j+ 1 N 1 K s s + s + s h C K s s + s + s h 2 C 2 j 1 j empt graph 11

1D Ising Model H N 1 N E = = K s s h s + C k T B j j + 1 j j = 1 j = 1 K K K K K K h h h h h h h exact solution periodic bc: here: h =, graphical solution ( ) 2 2 Z = Tr e = e e { si =± 1} s T s = TrT ( ) H j 1 j j 1 j j j+ 1 j j+ 1 N 1 K s s + s + s h C K s s + s + s h 2 C 2 j 1 ( K s js j+ 1 ) = ( K ) + s js j+ 1 ( K ) exp cosh 1 tanh w w w w w w empt graph j 11

1D Ising Model H N 1 N E = = K s s h s + C k T B j j + 1 j j = 1 j = 1 K K K K K K h h h h h h h exact solution periodic bc: ( ) 2 2 Z = Tr e = e e { si =± 1} s T s = TrT ( ) H j 1 j j 1 j j j+ 1 j j+ 1 N 1 K s s + s + s h C K s s + s + s h 2 C 2 j 1 j here: h =, graphical solution onl even powers of s j survive ( K s js j+ 1 ) = ( K ) + s js j+ 1 ( K ) exp cosh 1 tanh w s j w w w w w empt graph ( ) (pbc) N N 1 N 1 2 cosh N 1+ Z = K w 11

1D Ising Model Continued lim N F = ln[ 2cosh K ] smooth function of K = J/k T, B N no phase transition for T > cum G( j) si si + j = w j i i + w w w w w w j 1 2 J / kbt ξ ξ <, for all T > = ln w T e 12

1D Ising Model Continued lim N F = ln[ 2cosh K ] smooth function of K = J/k T, B N no phase transition for T > cum G( j) si si + j = w j i i + w w w w w w j 1 2 J / kbt ξ ξ <, for all T > = ln w T e ( K ) exp 2 χ = G( j) / kbt = k T T j = pseudo-transition at T = B χ 1 k B MF T k c B T / J 12

1D Ising Model Continued lim N F = ln[ 2cosh K ] smooth function of K = J/k T, B N no phase transition for T > cum G( j) si si + j = w j i i + w w w w w w j 1 2 J / kbt ξ ln w e, for all T = T ( K ) exp 2 χ = G( j) / kbt = k T T j = pseudo-transition at T = B ξ < > χ 1 k B RG-> exponential increase of ξ is characteristic of sstems at lcd MF T k c B T / J 12

Decimation K K K K K K K K trace out black spins w = w b and C ( ) ( ) K = f K artanh tanh b K b RG flow for 1D Ising model w = 1 w = C T = T = 13

Decimation K K K K K K K K trace out black spins w = w b and C ( ) ( ) K = f K artanh tanh b K b RG flow for 1D Ising model w = 1 w = C T = T = dl b = e, dl, w w ( l) dw( l) dl = w( l)ln w( l) dk dl ( l) 1 sinh 2 K ln tanh ( K ) = 2 13

Exploiting the Flow Equation τ = 1/ K dτ ( l) dl τ 2 2 ; τ, no term linear in t on rhs! l dl l 2 2 2 = = + τ τ τ ( l ) exp( l ) ( l ) exp( 2 ) ˆ ξ = ˆ ξ ˆ ξ τ exponential increase of correlation length! 2D O(n) models, nonlinear σ model: dτ ( l) dl n τ τ 2 ( 2) ;, 14

Migdal-Kadanoff Renormalization Scheme a) move bonds: H H H = + H with H = 15

Migdal-Kadanoff Renormalization Scheme a) move bonds: H H H = + H with H = K = b K 2 2 K = K 1 1 15

Migdal-Kadanoff Renormalization Scheme a) move bonds: H H H = + H with H = F F lower bound! K = b K 2 2 K = K 1 1 15

Migdal-Kadanoff Renormalization Scheme a) move bonds: H H H = + H with H = F F lower bound! K = b K 2 2 K = K 1 1 b) trace out spins: ( ) ( ) b K 1 = artanh tanh K 1 fb K1 K = K 2 2 15

Migdal-Kadanoff Renormalization Scheme Continued ( d ) Result: ( K, K ) ( K, K ) = R ( K, K ) 1 2 1 2 b, 1 1 2 c) repeat for other directions 2,, d: R R R R R ( ) ( ) ( ) ( ) ( ) b b, d b, d 1 b,2 b,1 ( d ) d 1 Result: R ( K ) = b f ( K ) R b 1 b 1 ( ) ( ) K = b f b K ; j = 2,, d ( d ) d j j 1 b j b j f b artanh tanh b 16

Migdal-Kadanoff Flow Equations dl dk( l) 1 b = e, dl, = ( d 1) K + sinh 2 ln tanh ( ) 2 K K dl β ( d, K) K K*= K c β K ( 2, K ) T = β K ( 1, K ) K β K ( 1/ 2, K ) 17

Migdal-Kadanoff Flow Equations dl dk( l) 1 b = e, dl, = ( d 1) K + sinh 2 ln tanh ( ) 2 K K dl β ( d, K) K d [ K ] = 2 : sinh = 1 1 K = ln 1 + 2 c 2 exact! c T = K*= K c β K β K ( 2, K ) ( 1, K ) K reason: MK transform. commutes with dualit transformation! β K ( 1/ 2, K ) 17

ac H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theor Statistical Landau Theor (Landau, Ginzburg, Wilson) start with microscopic model: divide sstem into cells and coarse grain a Z = exp H { s } i { } { } { M } micro [ s ] exp [ ], = Hmicro si δ M c s j M s c j c c i = exp H c Hmicro meso [ s ] i [ M ] [ ] ([ ], ) ([ ], ) c Hmeso M c E M c T kbt S M c T h M c c = + continuum approximation: M c M c Cφ( x) + ( φ terms) i 18

Z H H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theor Mesoscopic Model = configurations [ φ ] ( φ ) D[ φ]exp H[ ] uv cutoff: Λ 2 π / ac H 1 τ u φ = φ + φ + φ hφ 2 2 4! d [ ] ( ) 2 2 4 d x V µ dimensions: [ φ] = ( d 2) / 2 2 [ τ ] = µ [ u ] = µ ε dimensionless interaction constant: u ε / 2 τ RG: e.g. Wilson s momentum shell scheme or field theor 19

Field Theor: Heuristic Considerations Λ (, ) ( x + x ) ( x ) (2) G x T φ φ cum regularized cumulants expect: G ( x, T ) x (2) ( d 2 + η ) Λ c but: G = length (2) ( d 2) Λ η ( ) ( Λ) (2) ( d 2) ϑ GΛ ( x, T ) = Λ 1+ + c C x x x ϑ > idea: limit Λ to extract asmptotic large- x behavior limit cannot be taken naivel! a) cut-off (to avoid uv divergences) reason: double role of Λ : -1 b) Λ = sole length remaining at Tc 2

Heuristic Intro To Renormalization trick: ( µ Λ) (2) η (2) Λ c µ Λ G,ren ( x, T, ) G ( x, T ) µ : arbitrar momentum scale η ( µ ) ( Λ) (2) ( d 2) ϑ GΛ,ren ( x, T, ) = 1+ + c µ C x x x c 21

Heuristic Intro To Renormalization trick: ( µ Λ) (2) η (2) Λ c µ Λ G,ren ( x, T, ) G ( x, T ) µ : arbitrar momentum scale η ( µ ) ( Λ) (2) ( d 2) ϑ GΛ,ren ( x, T, ) = 1+ + c µ C x x x Λ c Gren ( x, T, µ ) = C µ x (2) η ( d 2 + η ) c uv finite renormalized function! G ( x, T, µ ) = φ ( x + x ) φ ( x ) (2) ren ren ren c cum with ren ( ) 1/ 2 φ x Z φ ( x ), Z ( µ Λ) ( µ Λ) φ amplitude renormalization φ η 21

UV Divergences d 1 τ u H= d x φ φ φ 2 2 4! G ( N ) ( ) 2 2 + + 4 ( x,, x ) φ( x) φ( x ) 1 N 1 1 Gɶ ɶ q N cum ɶ ( q q ) q -1 ( N ) d = FT G 1,, N (2 π) δ j j ( q) Γ ( q) = + τ Σ( q) (2) (2) 2 2 = q + τ+ + + + 22

UV Divergences d 1 τ u H= d x φ φ φ 2 2 4! G ( N ) ( ) 2 2 + + 4 ( x,, x ) φ( x) φ( x ) 1 N 1 1 Gɶ ɶ q N ɶ cum ( ) -1 ( N ) d = FT G q1,, qn (2 π) δ q j j ( q) Γ ( q) = + τ Σ( q) (2) (2) 2 2 = q + τ+ + + + u d q Λ + C τ Λ q C n r d = 4 d d 2 d 4 1 d = d 2 2 2 q Λ (2 π ) + τ Λ + 4 τ l Λ, fo d 4 Λ ln Λ, for d = 4 22

UV Divergences d 1 τ u H= d x φ φ φ 2 2 4! G ( N ) ( ) 2 2 + + 4 ( x,, x ) φ( x) φ( x ) 1 N 1 1 Gɶ ɶ q N ɶ cum ( ) -1 ( N ) d = FT G q1,, qn (2 π) δ q j j ( q) Γ ( q) = + τ Σ( q) (2) (2) 2 2 = q + τ+ + + + q 2 ln Λ divergence u d q Λ + C τ Λ q C n r d = 4 d d 2 d 4 1 d = d 2 2 2 q Λ (2 π ) + τ Λ + 4 τ l Λ, fo d 4 Λ ln Λ, for d = 4 22

φ H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theor Renormalization d 1 τ u H= d x φ φ φ 2 2 4! 1/ 2 ren ( ) = Z φ ( ) 2 τ τ, c µ Zττ u φ = + ε = µ Z u u ( ) 2 2 + + 4 x x amplitude temperature ( mass ) coupling constant Z, Z, Z φ τ u uv divergent ln Λ for d = 4 = uv finite for d < 4 τ 2 Λ d = for 4 Λ for d < 4, c d 2 φ 4 theor: d 4 23

φ H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theor Renormalization d 1 τ u H= d x φ φ φ 2 2 4! 1/ 2 ren ( ) = Z φ ( ) 2 τ = τ, c + µ Zττ u φ ε = µ Z u u ( ) 2 2 + + 4 x x amplitude temperature ( mass ) coupling constant Z, Z, Z φ τ u uv divergent ln Λ for d = 4 = uv finite for d < 4 τ 2 Λ d = for 4 Λ for d < 4, c d 2 theorem (Bogoliubov, Parasiuk, Hepp, Zimmermann) for renormalizable theories: At an order of perturbation theor all uv singularities can be absorbed b a finite # of counterterms ( Z, Z, Z and ) such that the G are uv finite. ( N ) φ τ u τ, c ren φ 4 theor: d 4 23

RG Equations bare cumulants: independent of µ d dµ G ( x; τ, u ) ( N ) Λ = N / 2 N ( ) x τ ( µ ) ( τ ) ( N ) ( ) Gren ( x; τ, u, µ, Λ) = Zφ u, µ Λ GΛ ; τ, u,, Λ, u, u, µ, Λ µ beta fun µ 24

RG Equations bare cumulants: independent of µ d dµ G ( x; τ, u ) ( N ) Λ = N / 2 N ( ) x τ ( µ ) ( τ ) ( N ) ( ) G ren ( x ; τ, u, µ, Λ ) = Zφ u, µ Λ G Λ ; τ, u,, Λ, u, u, µ, Λ RGE: N ( N ) µ µ + βu u + ( 2 + ητ ) τ τ + ηφ ren ( ;,, ) = 2 G x u τ µ µ beta function: µ exponent functions : βu ( u, ε ) = µ u µ η ( u) = µ ln Z φ µ φ η ( u) = µ ln Z τ µ τ 24

Scale Invariance at Fixed Points assumption: ( ε u ) u * such that β, * = u RGE: N ( N ) µ µ + βu u + ( 2 + ητ ) τ τ + ηφ ren ( ;,, ) = 2 G x u τ µ ( d 2 η) ( ) (2) ( ;,, ) η + (2) ν ; G ren x u τ µ = µ x Ξ µ x τ u scale invariance for u = u*! 25

Scale Invariance at Fixed Points assumption: ( ε u ) u * such that β, * = u N ( N ) RGE: µ µ + βu u + ( 2 + η * τ ) τ τ + η * φ ren ( ;,, ) = 2 G x u τ µ ( d 2 η) ( ) (2) ( ;,, ) η + (2) ν ; G ren x u τ µ = µ x Ξ µ x τ u scale invariance for u = u*! 25

Scale Invariance at Fixed Points assumption: ( ε u ) u * such that β, * = u N ( N ) RGE: µ µ + βu u + ( 2 + η * τ ) τ τ + η * φ ren ( ;,, ) = 2 G x u τ µ dn Nη / 2 ( ) ( ) ren ( ;, τ, µ ) = µ µ Ξ µ τ ; ( N ) dn ν G x u x x u dn ( d ) = 2 2 η = η φ ( η ) ν = 1 2 + τ ( d 2 η) ( ) 1/ξ (2) ( ;,, ) η + (2) ν ; G ren x u τ µ = µ x Ξ µ x τ u scale invariance for u = u*! 25

Beta Functions βu ε ε 4 d > ir-stable u* = ( ) O ε Gaussian fixed point for b u u * d > 4 ε = d = 4 d < 4 26

Characteristics ( b) µ µ = µ b d N ( N ) b + ηφ ( u ) ren ( ;,, ) = 2 G x u τ µ db flow equations: d b u ( b) = βu u ( b) db d b τ ( b) = 2 + η u ( b) τ b db { τ } ( ) ( 1) u b = = u τ ( b 1) = = τ u u ( b) u b ω ( u u ) ω = β u > u u u τ ( ) = [, ] τ τ b b E u u τ b E [ u, u] τ τ τ 1 ν = 2 + η τ τ nonuniversal scale factors (upon inclusion of h) h h h ( b) = b E [ u, u] τ b E [ u, u] h ( η ) ν = d + 2 φ 2 h h h 27

Upshot η ( x;, τ, µ ) = [, ] ( ) G ren x;, τ, h, µ / G u b E u u G u b ( N ) N 2 ( N ) ren ( x u τ h µ ) dn Nη 2 ( N ) b E, G u u Gren ;,,, power of E h scaling function universalit (crit. expo s, scaling functions) two-scale factor universalit u ( ) ω corrections to scaling from terms b u u 28