Critical Behavior II: Renormalization Group Theory · H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory 2 What the Theory should Accomplish • Theory should

3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004

H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory

1

H. W. DiehlFachbereich Physik, UniversitätDuisburg-Essen, Campus Essen

Critical Behavior II:Renormalization Group Theory



2

What the Theory should Accomplish• Theory should yield & explain:

– scaling laws

– # of independent critical exponents

– scaling laws

– universality, two-scale-factor universality

– determinants for universality classes

– clarify to which universality class given microscopic system belongs

– numerically accurate, experimentally testable predictions

– crossover phenomena

– corrections to asymptotic behavior



3

RG Strategy

ˆ /≡ aξ ξ( )

ˆ ˆ

′ =

′ =

a a b

b

ξ ξ

ξ ξ

a' =a ba

′ ′→ = =ξ ξ increase length such that minimal a a ba

ˆ : ξlarge pert. theory fails ˆ : ′ξsmall pert. theory works

ξ ξ

additionalinteraction constants!



3

RG Strategy

ˆ /≡ aξ ξ( )

ˆ ˆ

′ =

′ =

a a b

b

ξ ξ

ξ ξ

a' =a ba



ξ ξ

( , )= ijK hK ( , )′ ′ ′= ijK hKadditionalinteraction constants!



3

RG Strategy

ˆ /≡ aξ ξ( )

ˆ ˆ

′ =

′ =

a a b

b

ξ ξ

ξ ξ

a' =a ba



ξ ξ

( , )= ijK hK ( , )′ ′ ′= ijK hKadditionalinteraction constants!



4

Recursion Relations

[ ]( )′→ =R bK K K

[ ] [ ] [ ]1ˆ ˆ ˆ ˆ−′ ′→ ≡ = bξ ξ ξ ξK K K

• fixed point: ( ): ∗ ∗ ∗ = RbK K K

( ) ( ') ( ')=�R R Rb b bb• important property:

* 1 *ˆ ˆ ˆ−′ ≡ = bξ ξ ξK K0 , or 0ˆ

, critical fixed point∗ = ∞

= ∞

Tξ K



5

RG Flow: 2D Ising-Model

B B/ ; /= =x x y yK J k T K J k T

1y

y

K

K+

1x

x

K

K+

0T =

T = ∞

/ 2=y xJ J

2=y xJ J



5



1y

y

K

K+

1x

x

K

K+

0T =

T = ∞

/ 2=y xJ J

2=y xJ J



5



1y

y

K

K+

1x

x

K

K+

0T =

T = ∞

/ 2=y xJ J

2=y xJ J



5



1y

y

K

K+

1x

x

K

K+

0T =

T = ∞

/ 2=y xJ J

2=y xJ J

/ 2=y xJ J

2=y xJ J



5



1y

y

K

K+

1x

x

K

K+

0T =

T = ∞

/ 2=y xJ J

2=y xJ J

/ 2=y xJ J

2=y xJ J

0=gτ

0=ig



6

Schematic RG Flows in a high dimensional space

unstable direction

stable manifold all points on

this stable basin of attraction flow to the fixed point



7

Linearization[ ]( )′→ =R bK K K

*= +δK K K

( )* ( ) *

( ) * 2

+ = +

= + ⋅ +

R

R

b

b O

δ δ

δ δ

K K K K

K K KL

( ) ( ) 1; −′= ≡ɶ uρ ρρ ρλ δ δU U U KL

= ⋅δ δK KL

( )*

∂ ≡ ∂

Rb

j

kKKL

not in general symmetric



7

Linearization[ ]( )′→ =R bK K K

*= +δK K K

( )* ( ) *

( ) * 2

+ = +

= + ⋅ +

R

R

b

b O

δ δ

δ δ

K K K K

K K KL

( ) ( ) 1; −′= ≡ɶ uρ ρρ ρλ δ δU U U KL

= ⋅δ δK KL

( )*

∂ ≡ ∂

Rb

j

kKKL

not in general symmetric

( ) : = ′→Rb u u uρ ρ ρ ρλ

RG eigenvaluelinear scaling field



8

RG Eigenexponents & Nonlinear Scaling Fields

( )( ) ( )

times

: ′→ =�…��R Rpb b

p

u u uρ ρ ρ ρλ

: = yy b ρ

ρ ρλRG eigenexponents

( ) ( ) ( )′ ′=�R R Rb b bb• important property:



8


( )( ) ( )

times

: ′→ =�…��R Rpb b

p

u u uρ ρ ρ ρλ

: = yy b ρ



( ) : = ′→Rb yu u b uρ

ρ ρ ρ

0 : :

0 : 0 :

0 : =

> → ±∞

< →

=

y u

y u

y u

ρ ρ

ρ ρ

ρ ρ

relevant

irrelevant

marginal

(+)



8


( )( ) ( )

times

: ′→ =�…��R Rpb b

p

u u uρ ρ ρ ρλ

: = yy b ρ



( ) : = ′→Rb yu u b uρ

ρ ρ ρ

0 : :

0 : 0 :

0 : =

> → ±∞

< →

=

y u

y u

y u

ρ ρ

ρ ρ

ρ ρ

relevant

irrelevant

marginal

(+)

( )′ ′′ ′ ′′= + +…g u C u uρ

ρ ρ ρ ρ ρ ρ

nonlinear scaling fields (Wegner): satisfy (+) even away from fixed pt.

“appropriate curvilinear coordinates”



9

Consequencesreduced free energy density:

∗= + δK K K

( ) ( ) ( )reg sing1 2, ,= + …f f f g gK K

( ) ( )1 2sing sing1 2 1 2, , , ,−=… …

y ydf g b bg b f g g

11choose such that 1 , 0 ,= ± ><yb b g gρ

( ) ( )1 2/sing sing1 2 1 2 1 1, , , 1, , ,

− −= ±… …id y

i if g g g g f g g g gϕ ϕ

0 if 0→ <iϕ1 : crossover exponent=i iy yϕ



9

Consequencesreduced free energy density:

∗= + δK K K

( ) ( ) ( )reg sing1 2, ,= + …f f f g gK K

( ) ( )1 2sing sing1 2 1 2, , , ,−=… …

y ydf g b bg b f g g

11choose such that 1 , 0 ,= ± ><yb b g gρ

( ) ( )1 2/sing sing1 2 1 2 1 1, , , 1, , ,

− −= ±… …id y

i if g g g g f g g g gϕ ϕ

0 if 0→ <iϕ1 : crossover exponent=i iy yϕ

( )/sing( , , ; , )−

±≈… …hd y

h i hf g g g g Y g gτ ϕτ τ τ

1 0,1 2 1,0; g≡ ≈ + + ≡ ≈ + +… …h

hg g c g cττ τ δµ δµ τ

( ) ( )singirrelevant1, ; 0± ± ==h hY g f g g

“dangerous irrelevant variables”

may be zero or !!∞



10

Scaling Operators

( ) ( )→ + ∫H H Odd x gρ ρδ x x

( )→ +g g gρ ρ ρδ x

( ) ( )∆ +=

∆ = −

…⋯ ⋯O Ob b

d y

ρρ ρ

ρ ρ

x x

2( )1 2 12 12

2( ) ( 2 )12 12 12

( ) ( ) ( ) ( / )

( )

− −

− − − − +

≡ =

=∼

h

c

h

d y

T

d y d

G x b G x b

G x x x η

φ φx x

Kadanoff, Patashinski & Pokrovskii



11

1D Ising Model1

11 1B

−

+= =

= = − − +∑ ∑N N

j j jj j

EK s s h s C

k TH

K K K K K K

h h h h h h h

( ){ }

( )1 1 1 12 2

1

1

1

2 2Tr

Tr

− − + ++ + − + + −−

=±

−

−

= =

=

∑ ��⋯ ⋯H j j j j j j j j

i

K s s s s h C K s s s s h

s

j j

C

N

Z e e e

s sT

T

exact solution

periodic bc:

empty graph



11

1D Ising Model1

11 1B

−

+= =

= = − − +∑ ∑N N

j j jj j

EK s s h s C

k TH

K K K K K K

h h h h h h h

( ){ }

( )1 1 1 12 2

1

1

1

2 2Tr

Tr

− − + ++ + − + + −−

=±

−

−

= =

=

∑ ��⋯ ⋯H j j j j j j j j

i


s

j j

C

N

Z e e e

s sT

T

exact solution

periodic bc:

here: h = 0, “graphical solution”

( ) ( ) ( )1 1exp cosh 1 tanh+ + = + j j j jK s s K s s K��

w

w w w w wempty graph



11

1D Ising Model1

11 1B

−

+= =

= = − − +∑ ∑N N

j j jj j

EK s s h s C

k TH

K K K K K K

h h h h h h h

( ){ }

( )1 1 1 12 2

1

1

1

2 2Tr

Tr

− − + ++ + − + + −−

=±

−

−

= =

=

∑ ��⋯ ⋯H j j j j j j j j

i


s

j j

C

N

Z e e e

s sT

T

exact solution

periodic bc:



w

w w w w w

( ) 1(pbc 1) 2 cosh 1− −+ =

NNNNZ K w

only powers

of survive ∑j

js

s

evenempty graph



11

1D Ising Model1

11 1B

−

+= =

= = − − +∑ ∑N N

j j jj j

EK s s h s C

k TH

K K K K K K

h h h h h h h

( ){ }

( )1 1 1 12 2

1

1

1

2 2Tr

Tr

− − + ++ + − + + −−

=±

−

−

= =

=

∑ ��⋯ ⋯H j j j j j j j j

i


s

j j

C

N

Z e e e

s sT

T

exact solution

periodic bc:



w

w w w w w

( ) 1(pbc 1) 2 cosh 1− −+ =

NNNNZ K w

only powers

of survive ∑j

js

s

even

empty graph



12

1D Ising Model Continued

[ ]lim ln 2cosh→∞

= −N

KN

F smooth function of K = J/kBT, no phase transition for T > 0

cum( ) +≡ = j

i i jG j s s ww w w w ww

i +i j

B2 /1

0ln −−

→= − ≈ J k T

Tw eξ , for all 0< ∞ >Tξ



12



= −N

KN


cum( ) +≡ = j


i +i j

B2 /1

0ln −−

→= − ≈ J k T


( )B

B

exp 2( ) /

∞

→∞=−∞

= = → ∞∑T

j

KG j k T

k Tχ

pseudo-transition at T = 0

MFcT B /k T J

1B

− kχ



12



= −N

KN


cum( ) +≡ = j


i +i j

B2 /1

0ln −−

→= − ≈ J k T


( )B

B

exp 2( ) /

∞

→∞=−∞

= = → ∞∑T

j

KG j k T

k Tχ

pseudo-transition at T = 0

MFcT B /k T J

1B

− kχ

RG-> exponential increase of ξ is characteristic of systems at lcd



13

Decimation

K K K K K K ′K ′K

trace out black spins ′ = bw w and ′ ≠C C

= ∞T0=T

1=w 0=wRG flow for 1D Ising model

( ) ( )artanh tanh′ = ≡ bbK f K K



13

Decimation

K K K K K K ′K ′K

trace out black spins ′ = bw w and ′ ≠C C

= ∞T0=T

1=w 0=wRG flow for 1D Ising model

( ) ( )artanh tanh′ = ≡ bbK f K K

, 0,= →dlb e dl ( )→ ℓw w ( )( ) ln ( )=ℓ ℓ ℓ

ℓ

dww w

d

( )( ) 1sinh 2 ln tanh

2 =

ℓ

ℓ

dKK K

d



14

Exploiting the Flow Equation

1/= Kτ2( )

; 0,2

≈ →ℓ

ℓ

d

d

τ τ τ

0

000

2 2 2= = − + ≈∫ℓ

ℓ ℓdτ τ τ ( ) ( ) ( ) ( )0 0 0

ˆ ˆ ˆexp exp 2= ≈ℓ ℓ ℓξ ξ ξ τ

exponential increase of correlation length!

2( )( 2) ; 0,− →ℓ∼

ℓ

dn

d

τ τ τ2D O(n) models, nonlinear σ model:

no term linear in t on rhs!



15

Migdal-Kadanoff Renormalization Scheme

a) move bonds: with 0′→ = + ∆ ∆ =H H�H H H



15



2 2′ =K b K

1 1′ =K K



15



2 2′ =K b K

1 1′ =K K

′≥F F

lower bound!



15



2 2′ =K b K

1 1′ =K K

′≥F F

lower bound!

b) trace out spins:

( ) ( )1 1 1artanh tanh′′ ′ ′ = ≡ b

bK K f K

2 2′′ ′=K K



16

Migdal-Kadanoff Renormalization Scheme Continued

Result: ( ) ( ) ( )( )1 2 1 2 11, 2, , ,′′ ′′ =֏

dbK K K K K KR

( ) ( ) ( ) ( ) ( ), , 1 ,2 ,1−≡ � �…� �

d d d d db b d b d b bR R R R R

c) repeat for other directions 2, …, d:

Result: ( ) ( )( ) ( )

( ) 11 1

( ) 1 ; 2, ,

artanh tanh

−

− −

=

= =

≡

⋯

�

R

R

d db b

d d j jb j b j

bb

K b f K

K b f b K j d

f



17

Migdal-Kadanoff Flow Equations

, 0,= →dlb e dl ( )( ) 1( 1) sinh 2 ln tan

2

( , )

h

−

= − + ��

ℓ

ℓ

K

dKd K

d K

K Kd

β

( )2,− K Kβ

( )1,− K Kβ

( )1/ 2,− K Kβ

K*= Kc

= ∞TK



17



2

( , )

h

−

= − + ��

ℓ

ℓ

K

dKd K

d K

K Kd

β

( )2,− K Kβ

( )1,− K Kβ

( )1/ 2,− K Kβ

K*= Kc

= ∞TK

[ ]2 : sinh 1

1ln 1 2

2

= ⇒ =

⇒ = +

c

c

d K

K

exact!

reason: MK transform. commutes with duality transformation!



17



2

( , )

h

−

= − + ��

ℓ

ℓ

K

dKd K

d K

K Kd

β

( )2,− K Kβ

( )1,− K Kβ

( )1/ 2,− K Kβ

K*= Kc

= ∞TK

[ ]2 : sinh 1

1ln 1 2

2

= ⇒ =

⇒ = +

c

c

d K

K

exact!

reason: MK transform. commutes with duality transformation!

11 , 1:

2= + ⇒ ≈≪ cd Kε ε

ε

integrate flow equations:

1/ 1/= ≈yτν ε



18

Statistical Landau Theory(Landau, Ginzburg, Wilson)

• divide system into cells and coarse grain

[ ]{ }

[ ]{ }{ }

[ ]{ }

micro

micro

meso

exp

exp ,

exp

∈

= −

= −

= −

∑

∑∑ ∑∏

∑

H

H

H

i

c i

c

is

i c jM s j cc

cM

Z s

s M s

M

δ≫Ca a

[ ] ( ) ( )meso B[ ], [ ],= − − ∑H c c c cc

M E M T k T S M T h M

+ continuum approximation: ( )( ) terms′− + ∇≃c cM M Cφ φx

[ ]microH is• start with microscopic model:



19

Mesoscopic Model

[ ] ( )2 2 40 01

2 2 4! = ∇ + + −

∫Hd

V

ud x h

τφ φ φ φ φ

( )configurations [ ]

[ ]exp [ ]= −∫HH��Z D

φ

φ φ

20

0

[ ] ( 2) / 2

[ ]

[ ]

= −=

=

d

u ε

φτ µ

µ

µ dimensions:

dimensionless interaction constant:/ 2

0 0−u ετ

uv cutoff: 2 /Λ ∼ caπ

RG: e.g. Wilson’s momentum shell scheme or field theory



20

Field Theory: Heuristic Considerations

cum(2)0 0( , ) ( ) ( )Λ ≡ +G x T φ φx x x

(2) ( 2 )( , ) − − +Λ ∼

dcG x T x η

expect:(2) ( 2)but: length− −Λ =

dG

( ) ( )(2) ( 2)( , ) 1− −− −

Λ

Λ= +Λ +…

dcG x T C x x x

η ϑ0>ϑ

regularized cumulants



20

Field Theory: Heuristic Considerations

cum(2)0 0( , ) ( ) ( )Λ ≡ +G x T φ φx x x

(2) ( 2 )( , ) − − +Λ ∼

dcG x T x η

expect:(2) ( 2)but: length− −Λ =

dG

( ) ( )(2) ( 2)( , ) 1− −− −

Λ

Λ= +Λ +…

dcG x T C x x x

η ϑ0>ϑ

regularized cumulants

idea: limit to extract asymptotic large- behaviorΛ → ∞ x

limit cannot be taken naively!

reason:-1

a) cut-off role of :

b) sole remaining at

Λ Λ = c

(to avoid uv divergences)double

length T



21

Heuristic Intro To Renormalization

( )(2) (2),ren( , , ) ( , )

−Λ Λ≡ Λc cG x T G x T

ηµµ

( ) ( )(2) ( 2),ren( , , ) 1

− −− −Λ

= + +Λ

…d

cG x T C x x xη ϑµµ

trick:

µ : arbitrary momentum scale



21


( )(2) (2),ren( , , ) ( , )


ηµµ

( ) ( )(2) ( 2),ren( , , ) 1

− −− −Λ

= + +Λ

…d


trick:


(2) ( 2 )ren ( , , ) − − +−= d

cG x T C x ηηµ µ

Λ → ∞

uv finite renormalized function!



21


( )(2) (2),ren( , , ) ( , )


ηµµ

( ) ( )(2) ( 2),ren( , , ) 1

− −− −Λ

= + +Λ

…d


trick:


(2) ( 2 )ren ( , , ) − − +−= d

cG x T C x ηηµ µ

Λ → ∞

uv finite renormalized function!

cum(2) ren renren 0 0( , , ) ( ) ( )= +cG x T µ φ φx x x

with ( ) ( )1/ 2ren( ) ( ) ,−≡ Λ Λ∼Z Zη

φ φφ φ µ µx x

amplitude renormalization



22

UV Divergences

( )2 420 0 1

2 2 4!dd x

uH φ φ φ

τ=

∇ + +

∫

( ) ( ) ( )

( )

cum( )1 1

-1 ( )1

, ,

FT , , (2 )

NN N

N dN j

j

G

G

x x x x

q q q

φ φ

π δ

≡ =

∑

… ⋯

ɶ …

( ) ( ) ( )(2) (2) 20

20

1 G q

q

q q qτ

τ

≡ Γ = + −Σ

= + + + + +

ɶ ɶ

…



22

UV Divergences

( )2 420 0 1

2 2 4!dd x

uH φ φ φ

τ=

∇ + +

∫

( ) ( ) ( )

( )

cum( )1 1

-1 ( )1

, ,

FT , , (2 )

NN N

N dN j

j

G

G

x x x x

q q q

φ φ

π δ

≡ =

∑

… ⋯

ɶ …

( ) ( ) ( )(2) (2) 20

20

1 G q

q

q q qτ

τ

≡ Γ = + −Σ

= + + + + +

ɶ ɶ

…

2 400

20

24 0

1

2 (2 ) , fo l rn 4

− −

≤Λ

Λ

Λ + Λ= − Λ+ + =∫ ∼

d ddd

dq

C

q C d

u d q τπ τ τ

4

, for ln 4

−

Λ =

Λ∼

d

d



22

UV Divergences

( )2 420 0 1

2 2 4!dd x

uH φ φ φ

τ=

∇ + +

∫

( ) ( ) ( )

( )

cum( )1 1

-1 ( )1

, ,

FT , , (2 )

NN N

N dN j

j

G

G

x x x x

q q q

φ φ

π δ

≡ =

∑

… ⋯

ɶ …

( ) ( ) ( )(2) (2) 20

20

1 G q

q

q q qτ

τ

≡ Γ = + −Σ

= + + + + +

ɶ ɶ

…

2 400

20

24 0

1

2 (2 ) , fo l rn 4

− −

≤Λ

Λ

Λ + Λ= − Λ+ + =∫ ∼

d ddd

dq

C

q C d

u d q τπ τ τ

4

, for ln 4

−

Λ =

Λ∼

d

d

q2 ln Λ divergence



23

Renormalization

( )2 420 0 1

2 2 4!dd x

uH φ φ φ

τ=

∇ + +

∫

( ) ( )1/ 2

0

re

20

0

,

n=

= +

=c

u

Z

Z

Zu uε

φ

τ

φ φ

τ µ τ

µ

τ

x x amplitude

temperature (“mass”)

coupling constant

uv divergent ln for 4, ,

uv finite for 4

Λ == <

∼u

dZ Z Z

dφ τ

2

0, 2

for 4

for 4−

Λ =

Λ <∼c d

d

dτ

4 theory:

4≤d

φ



23

Renormalization

( )2 420 0 1

2 2 4!dd x

uH φ φ φ

τ=

∇ + +

∫

( ) ( )1/ 2

0

re

20

0

,

n=

= +

=c

u

Z

Z

Zu uε

φ

τ

φ φ

τ µ τ

µ

τ

x x amplitude

temperature (“mass”)

coupling constant

uv divergent ln for 4, ,

uv finite for 4

Λ == <

∼u

dZ Z Z

dφ τ

2

0, 2

for 4

for 4−

Λ =

Λ <∼c d

d

dτ

theorem (Bogoliubov, Parasiuk, Hepp, Zimmermann) for renormalizable theories:

( )0, ren

At any order of perturbation theory all uv singularities

can be absorbed by a finite # of counterterms

( , , and ) such that the are uv finite.Nu cZ Z Z Gφ τ τ

4 theory:

4≤d

φ



24

RG Equations

( )0 0

0

( ; , ) 0Λ =NdG u

dτ

µx

bare cumulants: independent of µ

( ) ( ) ( )/ 2( ) ( )ren 0 0( ; , , ) , ; , , , , , , ,,

−

Λ = Λ Λ Λ Λ NN NG u Z u G u u uφµ µ τ µτ τµτx x

0∂µµ beta function:



24

RG Equations

( )0 0

0

( ; , ) 0Λ =NdG u

dτ

µx

bare cumulants: independent of µ

( ) ( ) ( )/ 2( ) ( )ren 0 0( ; , , ) , ; , , , , , , ,,

−


( ) ( )ren2 ( ; , , ) 0

2 ∂ + ∂ + + ∂ + =

Nu u

NG uµ τ τ φµ β η τ η τ µx

0( , ) = ∂u u uµβ ε µbeta function:

“exponent functions”:

0

0

( ) ln

( ) ln

= ∂

= ∂

u Z

u Z

φ µ φ

τ µ τ

η µ

η µ

RGE:

( ) ( ) (/ 2( ) ( )ren 0 0( ; , , ) , ; , , , , , , ,,

−


0∂µµ beta function:



25

Scale Invariance at Fixed Points

( )* such that , * 0∃ =uu uβ εassumption:

( ) ( )ren2 ( ; , , ) 0

2 ∂ + ∂ + + ∂ + =

Nu u

NG uµ τ τ φµ β η τ η τ µxRGE:

( ) ( )2(2) (2)ren ( ; , , ) ;− − +∗ − ∗ = Ξ

dG u x x uηη ντ µ µ µ τx

scale invariance for u = u* !



25



( ) ( )ren2 ( ; , , ) 0

2 ∂ + ∂ + + ∂ + =

Nu u

NG uµ τ τ φµ β η τ η τ µxRGE: * *

( ) ( )2(2) (2)ren ( ; , , ) ;− − +∗ − ∗ = Ξ





25



( ) ( )ren2 ( ; , , ) 0

2 ∂ + ∂ + + ∂ + =

Nu u


( ) ( )/ 2( )ren ( ; , , ) ;

−∗ ∗ = Ξ NN

d NdNG u x x uη ντ µ µ µ µ τx

( )1 2 ∗= + τν η( )2 2= − −Nd d 1/ξ∗= φη η

( ) ( )2(2) (2)ren ( ; , , ) ;− − +∗ − ∗ = Ξ





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( ) ( )ren2 ( ; , , ) 0

2 ∂ + ∂ + + ∂ + =

Nu u


( ) ( )/ 2( )ren ( ; , , ) ;

−∗ ∗ = Ξ NN

d NdNG u x x uη ντ µ µ µ µ τx

( )1 2 ∗= + τν η( )2 2= − −Nd d 1/ξ∗= φη η

nontrivial fixed points? What if ? (generic case)∗∃ ≠u u

( ) ( )2(2) (2)ren ( ; , , ) ;− − +∗ − ∗ = Ξ





26

Beta Functionsuβ ε

* →u u

4 0≡ − >dε( )* =u O ε

for → ∞bGaussian fixed point

ir-stable

4<d0=ε4>d

4=d



27

Characteristics

( )→ =b bµ µ µ

( ) ( )

( ) ( ){ } ( )2

− =

= +

u

db u b u b

dbd

b b u b bdb τ

β

τ η τ

( ) ( )( )ren ; , , 0

2 − + =

Nd Nb u G u

db φη τ µx

( )

( )

1

1

= =

= =

u b u

bτ τflow equations:

( ) ( )−∗ ∗− −∼ uu b u b u uω 0∗= ∂ ∂ >u u uuω β

( ) [ , ]

1

]

2

[ ,∗

∗

= ≈

≡ = +

y y Eb b u uE u u b

y

τ τ

τ

τ

τ

ττ τ τν η

( )( ) [ , ]

2 2

[ , ]∗

∗= ≈

≡ ∆ = + −

h hh

y

h

yhh b b E u u b E u u h

y d φ

τ

ν η(upon inclusion of h)

nonuniversalscale factors



28

Upshot( ) [ ] ( )

( )( ) 2 ( )ren ren

2 ( )ren

; , , , ; , , , /

, ; , , ,

−

− − ∗ ∗

=

≈ N

N N NG

d N NG

G u b E u u G u b

b E u

h

hu G u

η

η

τ µ τ µ

τ µ

x x

x

power of Ehscaling function

• universality (crit. expo’s, scaling functions)

• two-scale factor universality

( )corrections to scaling from terms − ∗• −∼ ub u uω



28

Upshot( ) [ ] ( )

( )( ) 2 ( )ren ren

2 ( )ren

; , , , ; , , , /

, ; , , ,

−

− − ∗ ∗

=

≈ N

N N NG

d N NG

G u b E u u G u b

b E u

h

hu G u

η

η

τ µ τ µ

τ µ

x x

x

power of Ehscaling function

• universality (crit. expo’s, scaling functions)

• two-scale factor universality

( )corrections to scaling from terms − ∗• −∼ ub u uω

spatial isotropy + short-range interactions + scale invariance

-> conformal invariance! (Polyakov, Belavin, Zamolodchikov, Cardy…)

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