Critical Behavior II: Renormalization Group Theory · H. W. Diehl (Essen): Critical Behavior II:...
Transcript of Critical Behavior II: Renormalization Group Theory · H. W. Diehl (Essen): Critical Behavior II:...
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
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H. W. DiehlFachbereich Physik, UniversitätDuisburg-Essen, Campus Essen
Critical Behavior II:Renormalization Group Theory
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
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
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
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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!
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
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
( , )= ijK hK ( , )′ ′ ′= ijK hKadditionalinteraction constants!
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
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
( , )= ijK hK ( , )′ ′ ′= ijK hKadditionalinteraction constants!
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
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
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
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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
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
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
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
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
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
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
/ 2=y xJ J
2=y xJ J
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
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
/ 2=y xJ J
2=y xJ J
0=gτ
0=ig
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
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
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
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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
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
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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
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
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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:
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
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:
( ) : = ′→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
(+)
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
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:
( ) : = ′→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”
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
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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
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
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 !!∞
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
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
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
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
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
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:
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
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
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:
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 w
( ) 1(pbc 1) 2 cosh 1− −+ =
NNNNZ K w
only powers
of survive ∑j
js
s
evenempty graph
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
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:
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 w
( ) 1(pbc 1) 2 cosh 1− −+ =
NNNNZ K w
only powers
of survive ∑j
js
s
even
empty graph
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
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
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
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
( )B
B
exp 2( ) /
∞
→∞=−∞
= = → ∞∑T
j
KG j k T
k Tχ
pseudo-transition at T = 0
MFcT B /k T J
1B
− kχ
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
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
( )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
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
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
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
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
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
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!
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
15
Migdal-Kadanoff Renormalization Scheme
a) move bonds: with 0′→ = + ∆ ∆ =H H�H H H
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
15
Migdal-Kadanoff Renormalization Scheme
a) move bonds: with 0′→ = + ∆ ∆ =H H�H H H
2 2′ =K b K
1 1′ =K K
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
15
Migdal-Kadanoff Renormalization Scheme
a) move bonds: with 0′→ = + ∆ ∆ =H H�H H H
2 2′ =K b K
1 1′ =K K
′≥F F
lower bound!
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
15
Migdal-Kadanoff Renormalization Scheme
a) move bonds: with 0′→ = + ∆ ∆ =H H�H H H
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
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
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
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
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
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
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
[ ]2 : sinh 1
1ln 1 2
2
= ⇒ =
⇒ = +
c
c
d K
K
exact!
reason: MK transform. commutes with duality transformation!
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
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
[ ]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
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
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:
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
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
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
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
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
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
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
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
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
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
(2) ( 2 )ren ( , , ) − − +−= d
cG x T C x ηηµ µ
Λ → ∞
uv finite renormalized function!
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
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
(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
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
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
…
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
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
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
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
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
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
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
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
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
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:
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
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
( ) ( )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( ; , , ) , ; , , , , , , ,,
−
Λ = Λ Λ Λ Λ NN NG u Z u G u u uφµ µ τ µτ τµτx x
0∂µµ beta function:
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
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* !
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
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* !
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
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( )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
dG u x x uηη ντ µ µ µ τx
scale invariance for u = u* !
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
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( )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
dG u x x uηη ντ µ µ µ τx
scale invariance for u = u* !
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
26
Beta Functionsuβ ε
* →u u
4 0≡ − >dε( )* =u O ε
for → ∞bGaussian fixed point
ir-stable
4<d0=ε4>d
4=d
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
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
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
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
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
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…)