Critical Behavior II: Renormalization Group Theory
|
|
- Everett Miles
- 6 years ago
- Views:
Transcription
1 Critical Behavior II: Renormalization Group Theor H. W. Diehl Fachbereich Phsik, Universität Duisburg-Essen, Campus Essen 1
2 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
3 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
4 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
5 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
6 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
7 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
8 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
9 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
10 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
11 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
12 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
13 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
14 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
15 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
16 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 8
17 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
18 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
19 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
20 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 c 2( d h ) ( d 2 + η ) = 12 G( x ) x x 1
21 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
22 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
23 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
24 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
25 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
26 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
27 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
28 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
29 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
30 Exploiting the Flow Equation τ = 1/ K dτ ( l) dl τ 2 2 ; τ, no term linear in t on rhs! l dl l = = + 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
31 Migdal-Kadanoff Renormalization Scheme a) move bonds: H H H = + H with H = 15
32 Migdal-Kadanoff Renormalization Scheme a) move bonds: H H H = + H with H = K = b K 2 2 K = K
33 Migdal-Kadanoff Renormalization Scheme a) move bonds: H H H = + H with H = F F lower bound! K = b K 2 2 K = K
34 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
35 Migdal-Kadanoff Renormalization Scheme Continued ( d ) Result: ( K, K ) ( K, K ) = R ( K, K ) b, 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
36 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
37 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 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
38 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 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 ) d = 1 + ε, ε 1: K c 1 2ε integrate flow equations: ν = 1/ 1/ ε τ 17
39 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
40 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 [ ] ( ) 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
41 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 > 2
42 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
43 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
44 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! 21
45 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
46 UV Divergences d 1 τ u H= d x φ φ φ 2 2 4! G ( N ) ( ) ( 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
47 UV Divergences d 1 τ u H= d x φ φ φ 2 2 4! G ( N ) ( ) ( 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 q Λ (2 π ) + τ Λ + 4 τ l Λ, fo d 4 Λ ln Λ, for d = 4 22
48 UV Divergences d 1 τ u H= d x φ φ φ 2 2 4! G ( N ) ( ) ( 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 q Λ (2 π ) + τ Λ + 4 τ l Λ, fo d 4 Λ ln Λ, for d = 4 22
49 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 ( ) 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
50 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 ( ) 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
51 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
52 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
53 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
54 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
55 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 = 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
56 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 = 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*! nontrivial fixed points? What if u u? (generic case) 25
57 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
58 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
59 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
60 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 spatial isotrop + short-range interactions + scale invariance -> conformal invariance! (Polakov, Belavin, Zamolodchikov, Card ) 28
Critical Behavior I: Phenomenology, Universality & Scaling
Critical Behavior I: Phenomenology, Universality & Scaling H. W. Diehl Fachbereich Physik, Universität Duisburg-Essen, Campus Essen 1 Goals recall basic facts about (static equilibrium) critical behavior
More informationThe Ising model Summary of L12
The Ising model Summary of L2 Aim: Study connections between macroscopic phenomena and the underlying microscopic world for a ferromagnet. How: Study the simplest possible model of a ferromagnet containing
More information8.334: Statistical Mechanics II Spring 2014 Test 2 Review Problems
8.334: Statistical Mechanics II Spring 014 Test Review Problems The test is closed book, but if you wish you may bring a one-sided sheet of formulas. The intent of this sheet is as a reminder of important
More informationPhysics 212: Statistical mechanics II Lecture XI
Physics 212: Statistical mechanics II Lecture XI The main result of the last lecture was a calculation of the averaged magnetization in mean-field theory in Fourier space when the spin at the origin is
More informationCritical Region of the QCD Phase Transition
Critical Region of the QCD Phase Transition Mean field vs. Renormalization group B.-J. Schaefer 1 and J. Wambach 1,2 1 Institut für Kernphysik TU Darmstadt 2 GSI Darmstadt 18th August 25 Uni. Graz B.-J.
More informationthe renormalization group (RG) idea
the renormalization group (RG) idea Block Spin Partition function Z =Tr s e H. block spin transformation (majority rule) T (s, if s i ; s,...,s 9 )= s i > 0; 0, otherwise. b Block Spin (block-)transformed
More informationPhase Transitions and the Renormalization Group
School of Science International Summer School on Topological and Symmetry-Broken Phases Phase Transitions and the Renormalization Group An Introduction Dietmar Lehmann Institute of Theoretical Physics,
More informationCritical exponents in quantum Einstein gravity
Critical exponents in quantum Einstein gravity Sándor Nagy Department of Theoretical physics, University of Debrecen MTA-DE Particle Physics Research Group, Debrecen Leibnitz, 28 June Critical exponents
More informationRenormalization Group: non perturbative aspects and applications in statistical and solid state physics.
Renormalization Group: non perturbative aspects and applications in statistical and solid state physics. Bertrand Delamotte Saclay, march 3, 2009 Introduction Field theory: - infinitely many degrees of
More informationComplex Systems Methods 9. Critical Phenomena: The Renormalization Group
Complex Systems Methods 9. Critical Phenomena: The Renormalization Group Eckehard Olbrich e.olbrich@gmx.de http://personal-homepages.mis.mpg.de/olbrich/complex systems.html Potsdam WS 2007/08 Olbrich (Leipzig)
More informationRenormalization Group for the Two-Dimensional Ising Model
Chapter 8 Renormalization Group for the Two-Dimensional Ising Model The two-dimensional (2D) Ising model is arguably the most important in statistical physics. This special status is due to Lars Onsager
More informationNUMERICAL METHODS FOR QUANTUM IMPURITY MODELS
NUMERICAL METODS FOR QUANTUM IMPURITY MODELS http://www.staff.science.uu.nl/~mitch003/nrg.html March 2015 Andrew Mitchell, Utrecht University Quantum impurity problems Part 1: Quantum impurity problems
More informationIntroduction to Phase Transitions in Statistical Physics and Field Theory
Introduction to Phase Transitions in Statistical Physics and Field Theory Motivation Basic Concepts and Facts about Phase Transitions: Phase Transitions in Fluids and Magnets Thermodynamics and Statistical
More informationScaling Theory. Roger Herrigel Advisor: Helmut Katzgraber
Scaling Theory Roger Herrigel Advisor: Helmut Katzgraber 7.4.2007 Outline The scaling hypothesis Critical exponents The scaling hypothesis Derivation of the scaling relations Heuristic explanation Kadanoff
More informationKolloquium Universität Innsbruck October 13, The renormalization group: from the foundations to modern applications
Kolloquium Universität Innsbruck October 13, 2009 The renormalization group: from the foundations to modern applications Peter Kopietz, Universität Frankfurt 1.) Historical introduction: what is the RG?
More informationRenormalization of microscopic Hamiltonians. Renormalization Group without Field Theory
Renormalization of microscopic Hamiltonians Renormalization Group without Field Theory Alberto Parola Università dell Insubria (Como - Italy) Renormalization Group Universality Only dimensionality and
More informationPhase transitions and critical phenomena
Phase transitions and critical phenomena Classification of phase transitions. Discontinous (st order) transitions Summary week -5 st derivatives of thermodynamic potentials jump discontinously, e.g. (
More informationPhysics 127b: Statistical Mechanics. Renormalization Group: 1d Ising Model. Perturbation expansion
Physics 17b: Statistical Mechanics Renormalization Group: 1d Ising Model The ReNormalization Group (RNG) gives an understanding of scaling and universality, and provides various approximation schemes to
More informationLecture Notes, Field Theory in Condensed Matter Physics: Quantum Criticality and the Renormalization Group
Lecture Notes, Field Theory in Condensed Matter Physics: Quantum Criticality and the Renormalization Group Jörg Schmalian Institute for Theory of Condensed Matter (TKM) Karlsruhe Institute of Technology
More informationPhase transitions and finite-size scaling
Phase transitions and finite-size scaling Critical slowing down and cluster methods. Theory of phase transitions/ RNG Finite-size scaling Detailed treatment: Lectures on Phase Transitions and the Renormalization
More informationRunning Couplings in Topologically Massive Gravity
Running Couplings in Topologically Massive Gravity Roberto Percacci 1 Ergin Sezgin 2 1 SISSA, Trieste 2 Texas A& M University, College Station, TX ERG 2010 - Corfu arxiv:1002.2640 [hep-th] - C& QG 2010
More informationEntanglement entropy and the F theorem
Entanglement entropy and the F theorem Mathematical Sciences and research centre, Southampton June 9, 2016 H RESEARH ENT Introduction This talk will be about: 1. Entanglement entropy 2. The F theorem for
More informationarxiv:cond-mat/ v4 [cond-mat.stat-mech] 19 Jun 2007
arxiv:cond-mat/060065v4 [cond-mat.stat-mech] 9 Jun 007 Restoration of Isotropy in the Ising Model on the Sierpiński Gasket Naoto Yajima Graduate School of Human and Environmental Studies, Kyoto University,
More informationReal-Space Renormalisation
Real-Space Renormalisation LMU München June 23, 2009 Real-Space Renormalisation of the One-Dimensional Ising Model Ernst Ising Real-Space Renormalisation of the One-Dimensional Ising Model Ernst Ising
More informationChap.9 Fixed points and exponents
Chap.9 Fixed points and exponents Youjin Deng 09.1.0 The fixed point and its neighborhood A point in the parameter space which is invariant under will be called a fixed point R s R μ* = μ * s μ * λ = We
More informationarxiv:cond-mat/ v1 4 Aug 2003
Conductivity of thermally fluctuating superconductors in two dimensions Subir Sachdev arxiv:cond-mat/0308063 v1 4 Aug 2003 Abstract Department of Physics, Yale University, P.O. Box 208120, New Haven CT
More informationVI. Series Expansions
VI. Series Expansions VI.A Low-temperature expansions Lattice models can also be studied by series expansions. Such expansions start with certain exactly solvable limits, and typically represent perturbations
More informationRenormalization Group analysis of 2D Ising model
Renormalization Group analysis of D Ising model Amir Bar January 7, 013 1 Introduction In this tutorial we will see explicitly how RG can be used to probe the phase diagram of d > 1 systems, focusing as
More informationFERMION MASSES FROM A CROSSOVER MECHANISM
FERMION MASSES FROM A ROSSOVER MEHANISM Vincenzo ranchina and Emanuele Messina Department of Physics, University of atania, Italy Eleventh Workshop on Non-Perturbative Quantum hromodynamics June 6 -, 2,
More informationWilsonian and large N theories of quantum critical metals. Srinivas Raghu (Stanford)
Wilsonian and large N theories of quantum critical metals Srinivas Raghu (Stanford) Collaborators and References R. Mahajan, D. Ramirez, S. Kachru, and SR, PRB 88, 115116 (2013). A. Liam Fitzpatrick, S.
More information8.334: Statistical Mechanics II Problem Set # 4 Due: 4/9/14 Transfer Matrices & Position space renormalization
8.334: Statistical Mechanics II Problem Set # 4 Due: 4/9/14 Transfer Matrices & Position space renormalization This problem set is partly intended to introduce the transfer matrix method, which is used
More informationPHYSICAL REVIEW LETTERS
PHYSICAL REVIEW LETTERS VOLUME 76 4 MARCH 1996 NUMBER 10 Finite-Size Scaling and Universality above the Upper Critical Dimensionality Erik Luijten* and Henk W. J. Blöte Faculty of Applied Physics, Delft
More informationRenormalization Group Equations
Renormalization Group Equations Idea: study the dependence of correlation functions on the scale k Various options: k = Λ: Wilson, Wegner-Houghton, coarse graining constant physics, cutoff/uv insensitivity
More informationf(t,h) = t 2 g f (h/t ), (3.2)
Chapter 3 The Scaling Hypothesis Previously, we found that singular behaviour in the vicinity of a second order critical point was characterised by a set of critical exponents {α,β,γ,δ, }. These power
More informationAn introduction to lattice field theory
An introduction to lattice field theory, TIFR, Mumbai, India CBM School, Darjeeling 21 23 January, 2014 1 The path integral formulation 2 Field theory, divergences, renormalization 3 Example 1: the central
More informationarxiv:hep-th/ v2 1 Aug 2001
Universal amplitude ratios in the two-dimensional Ising model 1 arxiv:hep-th/9710019v2 1 Aug 2001 Gesualdo Delfino Laboratoire de Physique Théorique, Université de Montpellier II Pl. E. Bataillon, 34095
More information8.334: Statistical Mechanics II Spring 2014 Test 3 Review Problems
8.334: Statistical Mechanics II Spring 014 Test 3 Review Problems The test is closed book, but if you wish you may bring a one-sided sheet of formulas. The intent of this sheet is as a reminder of important
More informationLandau s Fermi Liquid Theory
Thors Hans Hansson Stockholm University Outline 1 Fermi Liquids Why, What, and How? Why Fermi liquids? What is a Fermi liquids? Fermi Liquids How? 2 Landau s Phenomenological Approach The free Fermi gas
More informationMCRG Flow for the Nonlinear Sigma Model
Raphael Flore,, Björn Wellegehausen, Andreas Wipf 23.03.2012 So you want to quantize gravity... RG Approach to QFT all information stored in correlation functions φ(x 0 )... φ(x n ) = N Dφ φ(x 0 )... φ(x
More informationLow dimensional interacting bosons
Low dimensional interacting bosons Serena Cenatiempo PhD defence Supervisors: E. Marinari and A. Giuliani Physics Department Sapienza, Università di Roma February 7, 2013 It was only in 1995 Anderson et
More informationIsing model and phase transitions
Chapter 5 Ising model and phase transitions 05 by Alessandro Codello 5. Equilibrium statistical mechanics Aspinsystemisdescribedbyplacingaspinvariableσ i {, } at every site i of a given lattice. A microstate
More informationThe square lattice Ising model on the rectangle
The square lattice Ising model on the rectangle Fred Hucht, Theoretische Physik, Universität Duisburg-Essen, 47048 Duisburg Introduction Connection to Casimir forces Free energy contributions Analytical
More informationNotes on Renormalization Group: Introduction
Notes on Renormalization Group: Introduction Yi Zou (Dated: November 19, 2015) We introduce brief istory for te development of renormalization group (RG) in pysics. Kadanoff s scaling ypoteis is present
More informationPhase Transitions and Renormalization:
Phase Transitions and Renormalization: Using quantum techniques to understand critical phenomena. Sean Pohorence Department of Applied Mathematics and Theoretical Physics University of Cambridge CAPS 2013
More informationVI.D Self Duality in the Two Dimensional Ising Model
VI.D Self Duality in the Two Dimensional Ising Model Kramers and Wannier discovered a hidden symmetry that relates the properties of the Ising model on the square lattice at low and high temperatures.
More informationRenormalization group in stochastic theory of developed turbulence 3
Renormalization group in stochastic theory of developed turbulence 3 p. 1/11 Renormalization group in stochastic theory of developed turbulence 3 Operator-product expansion and scaling in the inertial
More informationRenormalization group, Kondo effect and hierarchical models G.Benfatto, I.Jauslin & GG. 1-d lattice, fermions+impurity, Kondo problem.
Renormalization group, Kondo effect and hierarchical models G.Benfatto, I.Jauslin & GG 1-d lattice, fermions+impurity, Kondo problem H h = α=± ( L/2 1 x= L/2 H K =H 0 +λ α,α =± γ,γ =± ψ + α(x)( 1 2 1)ψ
More informationNon-perturbative beta-function in SU(2) lattice gauge fields thermodynamics
Non-perturbative beta-function in SU(2) lattice gauge fields thermodynamics O. Mogilevsky, N.N.Bogolyubov Institute for Theoretical Physics, National Academy of Sciences of Ukraine, 25243 Kiev, Ukraine
More informationAsymptotically safe Quantum Gravity. Nonperturbative renormalizability and fractal space-times
p. 1/2 Asymptotically safe Quantum Gravity Nonperturbative renormalizability and fractal space-times Frank Saueressig Institute for Theoretical Physics & Spinoza Institute Utrecht University Rapporteur
More informationNATURAL SCIENCES TRIPOS. Past questions. EXPERIMENTAL AND THEORETICAL PHYSICS Minor Topics. (27 February 2010)
NATURAL SCIENCES TRIPOS Part III Past questions EXPERIMENTAL AND THEORETICAL PHYSICS Minor Topics (27 February 21) 1 In one-dimension, the q-state Potts model is defined by the lattice Hamiltonian βh =
More informationLinear Theory of Evolution to an Unstable State
Chapter 2 Linear Theory of Evolution to an Unstable State c 2012 by William Klein, Harvey Gould, and Jan Tobochnik 1 October 2012 2.1 Introduction The simple theory of nucleation that we introduced in
More informationSHANGHAI JIAO TONG UNIVERSITY LECTURE
Lecture 7 SHANGHAI JIAO TONG UNIVERSITY LECTURE 7 017 Anthony J. Leggett Department of Physics University of Illinois at Urbana-Champaign, USA and Director, Center for Complex Physics Shanghai Jiao Tong
More informationScale without conformal invariance
Scale without conformal invariance Andy Stergiou Department of Physics, UCSD based on arxiv:1106.2540, 1107.3840, 1110.1634, 1202.4757 with Jean-François Fortin and Benjamín Grinstein Outline The physics:
More informationLoop optimization for tensor network renormalization
Yukawa Institute for Theoretical Physics, Kyoto University, Japan June, 6 Loop optimization for tensor network renormalization Shuo Yang! Perimeter Institute for Theoretical Physics, Waterloo, Canada Zheng-Cheng
More informationAsymptotically free nonabelian Higgs models. Holger Gies
Asymptotically free nonabelian Higgs models Holger Gies Friedrich-Schiller-Universität Jena & Helmholtz-Institut Jena & Luca Zambelli, PRD 92, 025016 (2015) [arxiv:1502.05907], arxiv:16xx.yyyyy Prologue:
More informationTHE HEATED LAMINAR VERTICAL JET IN A LIQUID WITH POWER-LAW TEMPERATURE DEPENDENCE OF DENSITY. V. A. Sharifulin.
THE HEATED LAMINAR VERTICAL JET IN A LIQUID WITH POWER-LAW TEMPERATURE DEPENDENCE OF DENSITY 1. Introduction V. A. Sharifulin Perm State Technical Universit, Perm, Russia e-mail: sharifulin@perm.ru Water
More informationStatistical Physics. Solutions Sheet 11.
Statistical Physics. Solutions Sheet. Exercise. HS 0 Prof. Manfred Sigrist Condensation and crystallization in the lattice gas model. The lattice gas model is obtained by dividing the volume V into microscopic
More informationPhase transitions beyond the Landau-Ginzburg theory
Phase transitions beyond the Landau-Ginzburg theory Yifei Shi 21 October 2014 1 Phase transitions and critical points 2 Laudau-Ginzburg theory 3 KT transition and vortices 4 Phase transitions beyond Laudau-Ginzburg
More informationAsymptotic safety in the sine Gordon model a
Asymptotic safety in the sine Gordon model a Sándor Nagy Department of Theoretical Physics, University of Debrecen Debrecen, February 23, 2015 a J. Kovács, S.N., K. Sailer, arxiv:1408.2680, to appear in
More informationIsing Model and Renormalization Group
Ising Model and Renormalization Group Felix Johannes Behrens April 24, 2018 Abstract This paper sums up the talk I gave in the Statistical Physics seminar by Prof. A. Mielke. The aim of this paper is to
More informationExact Functional Renormalization Group for Wetting Transitions in Dimensions.
EUROPHYSICS LETTERS Europhys. Lett., 11 (7), pp. 657-662 (1990) 1 April 1990 Exact Functional Renormalization Group for Wetting Transitions in 1 + 1 Dimensions. F. JULICHER, R. LIPOWSKY (*) and H. MULLER-KRUMBHAAR
More informationInterface Profiles in Field Theory
Florian König Institut für Theoretische Physik Universität Münster January 10, 2011 / Forschungsseminar Quantenfeldtheorie Outline φ 4 -Theory in Statistical Physics Critical Phenomena and Order Parameter
More informationClusters and Percolation
Chapter 6 Clusters and Percolation c 2012 by W. Klein, Harvey Gould, and Jan Tobochnik 5 November 2012 6.1 Introduction In this chapter we continue our investigation of nucleation near the spinodal. We
More informationRG Limit Cycles (Part I)
RG Limit Cycles (Part I) Andy Stergiou UC San Diego based on work with Jean-François Fortin and Benjamín Grinstein Outline The physics: Background and motivation New improved SE tensor and scale invariance
More informationTransverse Momentum Distributions of Partons in the Nucleon
Lattice 2008, Williamsburg 2008-07-18 Transverse Momentum Distributions of Partons in the Nucleon Bernhard Musch Technische Universität München presenting work in collaboration with LHPC and Philipp Hägler
More informationChapter 4. Renormalisation Group. 4.1 Conceptual Approach
Chapter 4 Renormalisation Group Previously, our analysis of the Ginzburg-Landau Hamiltonian revealed a formal breakdown of mean-field theory in dimensions below some upper critical dimension. Although
More informationEffective forces due to confined nearly critical fluctuations
Effective forces due to confined nearly critical fluctuations H. W. Diehl Fachbereich Physik, U. Duisburg-Essen June 24, 2009 Collaborators: Daniel Grüneberg, U. Duisburg-Essen F. M. Schmidt, U. Duisburg-Essen
More informationAsymptotic Safety in the ADM formalism of gravity
Asymptotic Safety in the ADM formalism of gravity Frank Saueressig Research Institute for Mathematics, Astrophysics and Particle Physics Radboud University Nijmegen J. Biemans, A. Platania and F. Saueressig,
More informationVI.D Self Duality in the Two Dimensional Ising Model
VI.D Self Duality in the Two Dimensional Ising Model Kramers and Wannier discovered a hidden symmetry that relates the properties of the Ising model on the square lattice at low and high temperatures.
More informationLinking U(2) U(2) to O(4) via decoupling
Linking U(2) U(2) to O(4) via decoupling Norikazu Yamada (KEK, GUAS) in collaboration with Tomomi Sato (KEK, GUAS) Based on Linking U(2) U(2) to O(4) model via decoupling, PRD91(2015)034025 [arxiv:1412.8026
More informationErnst Ising. Student of Wilhelm Lenz in Hamburg. PhD Thesis work on linear chains of coupled magnetic moments. This is known as the Ising model.
The Ising model Ernst Ising May 10, 1900 in Köln-May 11 1998 in Peoria (IL) Student of Wilhelm Lenz in Hamburg. PhD 1924. Thesis work on linear chains of coupled magnetic moments. This is known as the
More informationRenormalization group analysis for the 3D XY model. Francesca Pietracaprina
Renormalization group analysis for the 3D XY model Statistical Physics PhD Course Statistical physics of Cold atoms 5/04/2012 The XY model Duality transformation H = J i,j cos(ϑ i ϑ j ) Plan of the seminar:
More informationIII. The Scaling Hypothesis
III. The Scaling Hypothesis III.A The Homogeneity Assumption In the previous chapters, the singular behavior in the vicinity of a continuous transition was characterized by a set of critical exponents
More informationThe glass transition as a spin glass problem
The glass transition as a spin glass problem Mike Moore School of Physics and Astronomy, University of Manchester UBC 2007 Co-Authors: Joonhyun Yeo, Konkuk University Marco Tarzia, Saclay Mike Moore (Manchester)
More informationConformal Sigma Models in Three Dimensions
Conformal Sigma Models in Three Dimensions Takeshi Higashi, Kiyoshi Higashijima,, Etsuko Itou, Muneto Nitta 3 Department of Physics, Graduate School of Science, Osaka University, Toyonaka, Osaka 560-0043,
More informationarxiv: v1 [hep-th] 24 Mar 2016
Quantum-classical transition in the Caldeira-Leggett model J. Kovács a,b, B. Fazekas c, S. Nagy a, K. Sailer a a Department of Theoretical Physics, University of Debrecen, P.O. Box 5, H-41 Debrecen, Hungary
More informationField Theory Approach to Equilibrium Critical Phenomena
Field Theory Approach to Equilibrium Critical Phenomena Uwe C. Täuber Department of Physics (MC 0435), Virginia Tech Blacksburg, Virginia 24061, USA email: tauber@vt.edu http://www.phys.vt.edu/~tauber/utaeuber.html
More informationFermi Liquid and BCS Phase Transition
Fermi Liquid and BCS Phase Transition Yu, Zhenhua November 2, 25 Abstract Landau fermi liquid theory is introduced as a successful theory describing the low energy properties of most fermi systems. Besides
More informationRenormalization-group study of the replica action for the random field Ising model
arxiv:cond-mat/9906405v1 [cond-mat.stat-mech] 8 Jun 1999 Renormalization-group study of the replica action for the random field Ising model Hisamitsu Mukaida mukaida@saitama-med.ac.jp and Yoshinori Sakamoto
More informationImplications of Poincaré symmetry for thermal field theories
L. Giusti STRONGnet 23 Graz - September 23 p. /27 Implications of Poincaré symmetry for thermal field theories Leonardo Giusti University of Milano-Bicocca Based on: L. G. and H. B. Meyer JHEP 3 (23) 4,
More informationBayesian inverse problems with Laplacian noise
Bayesian inverse problems with Laplacian noise Remo Kretschmann Faculty of Mathematics, University of Duisburg-Essen Applied Inverse Problems 2017, M27 Hangzhou, 1 June 2017 1 / 33 Outline 1 Inverse heat
More informationDimensional Reduction in the Renormalisation Group:
Dimensional Reduction in the Renormalisation Group: From Scalar Fields to Quantum Einstein Gravity Natália Alkofer Advisor: Daniel Litim Co-advisor: Bernd-Jochen Schaefer 11/01/12 PhD Seminar 1 Outline
More informationUniversal scaling behavior of directed percolation and the pair contact process in an external field
Journal of Physics A 35, 005, (00) Universal scaling behavior of directed percolation and the pair contact process in an external field S. Lübeck, and R. D. Willmann,3 Weizmann Institute, Department of
More informationCrossover scaling in two dimensions
PHYSICAL REVIEW E VOLUME 56, NUMBER 6 DECEMBER 1997 Crossover scaling in two dimensions Erik Luijten * and Henk W. J. Blöte Department of Physics, Delft University of Technology, Lorentzweg 1, 2628 CJ
More informationTowards a quantitative FRG approach for the BCS-BEC crossover
Towards a quantitative FRG approach for the BCS-BEC crossover Michael M. Scherer Theoretisch Physikalisches Institut, Jena University in collaboration with Sebastian Diehl, Stefan Flörchinger, Holger Gies,
More informationQuantum phase transitions of insulators, superconductors and metals in two dimensions
Quantum phase transitions of insulators, superconductors and metals in two dimensions Talk online: sachdev.physics.harvard.edu HARVARD Outline 1. Phenomenology of the cuprate superconductors (and other
More informationNon-Fixed Point Renormalization Group Flow
Non-Fixed Point Renormalization Group Flow Gilberto de la Peña May 9, 2013 Abstract: The renormalization group flow of most systems is characterized by attractive or repelling fixed points. Nevertheless,
More informationTHE RENORMALIZATION GROUP AND WEYL-INVARIANCE
THE RENORMALIZATION GROUP AND WEYL-INVARIANCE Asymptotic Safety Seminar Series 2012 [ Based on arxiv:1210.3284 ] GIULIO D ODORICO with A. CODELLO, C. PAGANI and R. PERCACCI 1 Outline Weyl-invariance &
More information8.821 String Theory Fall 2008
MIT OpenCourseWare http://ocw.mit.edu 8.81 String Theory Fall 008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. 8.81 F008 Lecture 11: CFT continued;
More informationNewton s Method and Localization
Newton s Method and Localization Workshop on Analytical Aspects of Mathematical Physics John Imbrie May 30, 2013 Overview Diagonalizing the Hamiltonian is a goal in quantum theory. I would like to discuss
More informationin three-dimensional three-state random bond Potts model (α >0f for a disordered d dsystem in 3D)
Positive specific heat critical exponent in three-dimensional three-state random bond Potts model (α >0f for a disordered d dsystem in 3D) ZHONG Fan School of Physics and Engineering Sun Yat-sen University
More informationProblem set for the course Skálázás és renormálás a statisztikus fizikában, 2014
1 Problem set for the course Skálázás és renormálás a statisztikus fizikában, 014 Rules: You can choose at wish from problems having the same main number (i.e. from a given section), but you can collect
More information21 Renormalization group
Renormalization group. Renormalization and interpolation Probably because they describe point particles, quantum field theories are divergent. Unknown physics at very short distance scales, removes these
More informationDiffusive Transport Enhanced by Thermal Velocity Fluctuations
Diffusive Transport Enhanced by Thermal Velocity Fluctuations Aleksandar Donev 1 Courant Institute, New York University & Alejandro L. Garcia, San Jose State University John B. Bell, Lawrence Berkeley
More informationDilaton: Saving Conformal Symmetry
Dilaton: Saving Conformal Symmetry Alexander Monin Ecole Polytechnique Fédérale de Lausanne December 2, 2013 lexander Monin (Ecole Polytechnique Fédérale de Dilaton: Lausanne) Saving Conformal Symmetry
More informationLattice study of quantum entanglement in SU(3) Yang-Mills theory at zero and finite temperatures
Lattice study of quantum entanglement in SU(3) Yang-Mills theory at zero and finite temperatures Yoshiyuki Nakagawa Graduate School of Science and Technology, Niigata University, Igarashi-2, Nishi-ku,
More informationPhase Transitions in Condensed Matter Spontaneous Symmetry Breaking and Universality. Hans-Henning Klauss. Institut für Festkörperphysik TU Dresden
Phase Transitions in Condensed Matter Spontaneous Symmetry Breaking and Universality Hans-Henning Klauss Institut für Festkörperphysik TU Dresden 1 References [1] Stephen Blundell, Magnetism in Condensed
More informationThe TT Deformation of Quantum Field Theory
The TT Deformation of Quantum Field Theory John Cardy University of California, Berkeley All Souls College, Oxford ICMP, Montreal, July 2018 Introduction all QFTs that we use in physics are in some sense
More informationIntroduction to Renormalization Group
Introduction to Renormalization Group Alex Kovner University of Connecticut, Storrs, CT Valparaiso, December 12-14, 2013 Alex Kovner (UConn) Introduction to Renormalization Group December 12-14, 2013 1
More informationMott metal-insulator transition on compressible lattices
Mott metal-insulator transition on compressible lattices Markus Garst Universität zu Köln T p in collaboration with : Mario Zacharias (Köln) Lorenz Bartosch (Frankfurt) T c Mott insulator p c T metal pressure
More information