H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Critical Behavior II:
Renormalization Group Theory
H. W. Diehl
Fachbereich Physik, Universität
Duisburg-Essen, Campus Essen
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
1
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
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
2
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
RG Strategy
increase minimal length a → a ′ = ba such that ξ = ξ ′
additional
interaction
constants!
a ' = ba
a
ξ
ξˆ ≡ ξ / a
large ξˆ: pert. theory fails
ξ
ξ a ′ = (ξ a ) b
ξˆ′ = ξˆ b
small ξˆ′: pert. theory works
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
3
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
RG Strategy
increase minimal length a → a ′ = ba such that ξ = ξ ′
K = ( K ij , h)
K ′ = ( K ij′ , h′)
additional
interaction
constants!
a ' = ba
a
ξ
ξˆ ≡ ξ / a
large ξˆ: pert. theory fails
ξ
ξ a ′ = (ξ a ) b
ξˆ′ = ξˆ b
small ξˆ′: pert. theory works
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
3
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
RG Strategy
increase minimal length a → a ′ = ba such that ξ = ξ ′
K = ( K ij , h)
K ′ = ( K ij′ , h′)
additional
interaction
constants!
a ' = ba
a
ξ
ξˆ ≡ ξ / a
large ξˆ: pert. theory fails
ξ
ξ a ′ = (ξ a ) b
ξˆ′ = ξˆ b
small ξˆ′: pert. theory works
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
3
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Recursion Relations
K → K ′ = R (b ) [ K ]
ξˆ [ K ] → ξˆ′ ≡ ξˆ [ K ′] = b −1ξˆ [ K ]
• important property: R ( b ) R ( b ') = R ( bb ')
• fixed point:
ξˆ′ ≡ ξˆ  K *  = b −1 ξˆ  K * 
K ∗ : K ∗ = R ( b )  K ∗ 
0 , T = ∞ or 0
∗
ˆ
ξ  K  = 
∞ , critical fixed point
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
4
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
RG Flow: 2D Ising-Model
K x = J x / kBT ; K y = J y / kBT
T =0
Ky
1+ Ky
J y = 2J x
Jy = Jx / 2
T =∞
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
Kx
1+ Kx
5
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
RG Flow: 2D Ising-Model
K x = J x / kBT ; K y = J y / kBT
T =0
Ky
1+ Ky
J y = 2J x
Jy = Jx / 2
T =∞
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
Kx
1+ Kx
5
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
RG Flow: 2D Ising-Model
K x = J x / kBT ; K y = J y / kBT
T =0
Ky
1+ Ky
J y = 2J x
Jy = Jx / 2
T =∞
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
Kx
1+ Kx
5
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
RG Flow: 2D Ising-Model
K x = J x / kBT ; K y = J y / kBT
T =0
Ky
1+ Ky
J y = 2J x
Jy = Jx / 2
T =∞
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
Kx
1+ Kx
5
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
RG Flow: 2D Ising-Model
K x = J x / kBT ; K y = J y / kBT
T =0
Ky
J y = 2J x
1+ Ky
gτ = 0
gi = 0
Jy = Jx / 2
T =∞
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
Kx
1+ Kx
5
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Schematic RG Flows in a high
dimensional space
stable
manifold
unstable
direction
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
all points on
this stable
basin of
attraction
flow to the
fixed point
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H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Linearization
K → K ′ = R (b ) [ K ]
K = K* +δ K
K * + δ K = R ( b )  K * + δ K 
= R ( b )  K *  + L ⋅ δ K + O (δ K 2 )
not in general
symmetric
δ K = L ⋅δ K
Uɶ L U = ( λρ δ ρρ ′ ) ;
 ∂R j(b )

*
K 
L≡
 ∂K k   


−1
u
U
δK
≡
( ρ)
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
7
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Linearization
K → K ′ = R (b ) [ K ]
K = K* +δ K
K * + δ K = R ( b )  K * + δ K 
= R ( b )  K *  + L ⋅ δ K + O (δ K 2 )
not in general
symmetric
δ K = L ⋅δ K
Uɶ L U = ( λρ δ ρρ ′ ) ;
 ∂R j(b )

*
K 
L≡
 ∂K k   


−1
u
U
δK
≡
( ρ)
RG eigenvalue
linear scaling field
R ( b ) : u ρ → u′ρ = λρ u ρ
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
7
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
RG Eigenexponents & Nonlinear Scaling Fields
• important property: R
(b)
R
( b′ )
=R
( bb′ )
R
…
R : u ρ → u ′ρ = ( λρ ) u ρ
(b )
(b )
p
p times
RG eigenexponents y ρ : λρ = b
yρ
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
8
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
RG Eigenexponents & Nonlinear Scaling Fields
• important property: R
(b)
R
( b′ )
=R
( bb′ )
R
…
R : u ρ → u ′ρ = ( λρ ) u ρ
(b )
(b )
p
p times
RG eigenexponents y ρ : λρ = b
(+)
R
(b )
: u ρ → u′ρ = b u ρ
yρ
yρ
y ρ > 0 : u ρ → ±∞ : relevant
y ρ < 0 : u ρ → 0 : irrelevant
y ρ = 0 : u ρ = marginal
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
8
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
RG Eigenexponents & Nonlinear Scaling Fields
• important property: R
(b)
R
( b′ )
=R
( bb′ )
R
…
R : u ρ → u ′ρ = ( λρ ) u ρ
(b )
p
(b )
p times
RG eigenexponents y ρ : λρ = b
(+)
R
(b )
: u ρ → u′ρ = b u ρ
yρ
yρ
y ρ > 0 : u ρ → ±∞ : relevant
y ρ < 0 : u ρ → 0 : irrelevant
y ρ = 0 : u ρ = marginal
nonlinear scaling fields (Wegner):
satisfy (+) even away from fixed pt.
g ρ = u ρ + Cρ( ρ′ρ)′′ u ρ ′ u ρ ′′ + …
“appropriate curvilinear
coordinates”
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
8
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Consequences
f ( K ) = f reg ( K ) + f sing ( g1 , g 2 ,…)
reduced free energy density:
K = K∗ +δ K
(
f sing ( g1 , g 2 ,…) = b − d f sing b y1 g1 , b y2 g 2 ,…
choose b such that b g1
yρ
= ±1,
f sing ( g1 , g 2 ,… , gi ) = g1
)
g1 > 0,
<
d / y1
(
f sing ±1, g 2 g1
−ϕ 2
,… , gi g1
ϕ i = yi y1 : crossover exponent
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
−ϕ i
)
→ 0 if ϕi < 0
9
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Consequences
f ( K ) = f reg ( K ) + f sing ( g1 , g 2 ,…)
reduced free energy density:
K = K∗ +δ K
(
f sing ( g1 , g 2 ,…) = b − d f sing b y1 g1 , b y2 g 2 ,…
choose b such that b g1
yρ
= ±1,
f sing ( g1 , g 2 ,… , gi ) = g1
)
g1 > 0,
<
d / y1
(
f sing ±1, g 2 g1
−ϕ 2
,… , gi g1
ϕ i = yi y1 : crossover exponent
f sing ( gτ , g h ,…; gi ,…) ≈ gτ
d / yτ
(
Y± g h gτ
)
→ 0 if ϕi < 0
−ϕ h
τ
h
g1 ≡ gτ ≈ τ + c0,1
δµ + …; g 2 ≡ g h ≈ δµ + c1,0
τ +…
Y± ( g h ) = f sing ( ±1, g h ; girrelevant = 0 )
−ϕ i
)
may be zero or ∞ !!
“dangerous irrelevant
variables”
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
9
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Scaling Operators
g ρ → g ρ + δ g ρ ( x)
Kadanoff,
Patashinski &
Pokrovskii
H → H + ∫ d d x δ g ρ ( x ) Oρ ( x )
Oρ ( x b )⋯ = b
∆ ρ +…
Oρ ( x )⋯
∆ ρ = d − yρ
φ ( x1 ) φ ( x2 )
Tc
≡ G ( x12 ) = b −2( d − yh ) G ( x12 / b)
G ( x12 ) ∼ x12
−2( d − yh )
= x12
− ( d − 2 +η )
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
10
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
1D Ising Model
N −1
N
E
= − K ∑ s j s j +1 − h∑ s j + C
H=
kBT
j =1
j =1
exact solution
Z = Tr e − H =
periodic bc:
∑
{si =±1}
K K K K K K
h h h h h h h
Ks
(
s + s
+s
) h 2 −C 2
j −1 j
j −1
j
⋯ e
e
(
)
K s j s j +1 + s j + s j +1 h 2 − C 2
s j −1 T s j
= Tr T N −1
empty graph
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
11
⋯
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
1D Ising Model
N −1
N
E
= − K ∑ s j s j +1 − h∑ s j + C
H=
kBT
j =1
j =1
exact solution
Z = Tr e − H =
periodic bc:
∑
{si =±1}
K K K K K K
h h h h h h h
Ks
(
s + s
+s
) h 2 −C 2
j −1 j
j −1
j
⋯ e
e
(
)
K s j s j +1 + s j + s j +1 h 2 − C 2
s j −1 T s j
= Tr T N −1
here: h = 0,
“graphical
solution”
exp ( K s j s j +1 ) = cosh ( K ) 1 + s j s j +1 tanh ( K ) 
w
w w
w w w
empty graph
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
11
⋯
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
1D Ising Model
N −1
N
E
= − K ∑ s j s j +1 − h∑ s j + C
H=
kBT
j =1
j =1
exact solution
Z = Tr e − H =
periodic bc:
∑
K K K K K K
h h h h h h h
Ks
{si =±1}
(
s + s
+s
) h 2 −C 2
j −1 j
j −1
j
⋯ e
e
(
)
K s j s j +1 + s j + s j +1 h 2 − C 2
s j −1 T s j
= Tr T N −1
here: h = 0,
“graphical
solution”
exp ( K s j s j +1 ) = cosh ( K ) 1 + s j s j +1 tanh ( K ) 
w
w w
w w w
empty graph
only even powers
of s j survive ∑
Z
(pbc)
N
=2
N
( cosh K )
N −1
1 + w N −1 
sj
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
11
⋯
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
1D Ising Model
N −1
N
E
= − K ∑ s j s j +1 − h∑ s j + C
H=
kBT
j =1
j =1
exact solution
Z = Tr e − H =
periodic bc:
∑
K K K K K K
h h h h h h h
Ks
{si =±1}
(
s + s
+s
) h 2 −C 2
j −1 j
j −1
j
⋯ e
e
(
)
K s j s j +1 + s j + s j +1 h 2 − C 2
s j −1 T s j
= Tr T N −1
here: h = 0,
“graphical
solution”
exp ( K s j s j +1 ) = cosh ( K ) 1 + s j s j +1 tanh ( K ) 
w
w w
w w w
empty graph
only even powers
of s j survive ∑
Z
(pbc)
N
=2
N
( cosh K )
N −1
1 + w N −1 
sj
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
11
⋯
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
1D Ising Model Continued
F
= − ln [ 2 cosh K ]
N →∞ N
lim
G ( j ) ≡ si si + j
cum
=w
ξ −1 = − ln w ≈ e −2 J / k T
B
T →0
j
smooth function of K = J/kBT,
no phase transition for T > 0
i+ j
i
w w w w w w
ξ < ∞, for all T > 0
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
12
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
1D Ising Model Continued
F
= − ln [ 2 cosh K ]
N →∞ N
smooth function of K = J/kBT,
no phase transition for T > 0
lim
G ( j ) ≡ si si + j
cum
=w
w w w w w w
ξ < ∞, for all T > 0
ξ −1 = − ln w ≈ e −2 J / k T
B
T →0
χ=
∞
∑ G( j) / k T =
j =−∞
B
i+ j
i
j
exp ( 2 K )
k BT
χ −1 kB
→∞
T →∞
pseudo-transition at T = 0
TcMF
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
kBT / J
12
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
1D Ising Model Continued
F
= − ln [ 2 cosh K ]
N →∞ N
smooth function of K = J/kBT,
no phase transition for T > 0
lim
G ( j ) ≡ si si + j
cum
=w
w w w w w w
ξ < ∞, for all T > 0
ξ −1 = − ln w ≈ e −2 J / k T
B
T →0
χ=
∞
∑ G( j) / k T =
j =−∞
B
i+ j
i
j
exp ( 2 K )
k BT
χ −1 kB
→∞
T →∞
pseudo-transition at T = 0
RG-> exponential increase of ξ is
characteristic of systems at lcd
TcMF
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
kBT / J
12
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Decimation
K K K K K K
trace out black spins
K′
w′ = wb
K′
and C ′ ≠ C
K ′ = fb ( K ) ≡ artanh ( tanh b K )
RG flow for 1D Ising model
w =1
w=0
T =0
T =∞
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
13
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Decimation
K′
K K K K K K
w′ = wb
trace out black spins
K′
and C ′ ≠ C
K ′ = fb ( K ) ≡ artanh ( tanh b K )
RG flow for 1D Ising model
w =1
w=0
T =0
T =∞
b = e dl , dl → 0,
w → w(ℓ)
dw(ℓ)
= w(ℓ) ln w(ℓ)
dℓ
dK (ℓ) 1
= sinh  2 K  ln  tanh ( K ) 
dℓ
2
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
13
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Exploiting the Flow Equation
dτ (ℓ) τ 2
≈
; τ → 0,
dℓ
2
τ = 1/ K
∫
ℓ0
0
dℓ = ℓ0 = −
2
τ0
+
2
τ
≈
2
τ
no term linear in t
on rhs!
ξˆ = ξˆ ( ℓ 0 ) exp ( ℓ 0 ) ≈ ξˆ ( ℓ 0 ) exp ( 2 τ )
exponential increase of correlation length!
2D O(n) models,
nonlinear σ model:
dτ (ℓ)
∼ (n − 2)τ 2 ; τ → 0,
dℓ
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
14
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Migdal-Kadanoff Renormalization
Scheme
a) move bonds:
H → H′ = H + ∆H with
∆H = 0
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
15
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Migdal-Kadanoff Renormalization
Scheme
a) move bonds:
H → H′ = H + ∆H with
∆H = 0
K 2′ = b K 2
K1′ = K1
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
15
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Migdal-Kadanoff Renormalization
Scheme
a) move bonds:
H → H′ = H + ∆H with
∆H = 0
F ≥ F′
lower
bound!
K 2′ = b K 2
K1′ = K1
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
15
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Migdal-Kadanoff Renormalization
Scheme
a) move bonds:
H → H′ = H + ∆H with
∆H = 0
F ≥ F′
lower
bound!
K 2′ = b K 2
K1′ = K1
b) trace out spins:
K1′′ = artanh  tanh b ( K1′ )  ≡ fb ( K1′ )
K 2′′ = K 2′
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
15
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Migdal-Kadanoff Renormalization
Scheme Continued
Result:
( K1 , K 2 ) ֏ ( K1′′, K 2′′) = Rb(,d1 ) ( K1 , K 2 )
c) repeat for other directions 2, …, d:
Rb( d ) ≡ Rb(,dd) Rb(,dd)−1 … Rb(,2d ) Rb(,1d )
Result:
Rb( d ) ( K1 ) = b d −1 fb ( K1 )
Rb( d ) ( K j ) = b d − j fb ( b j −1 K j ) ; j = 2,⋯ , d
f b ≡ artanh tanh b
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
16
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Migdal-Kadanoff Flow Equations
b = e dl , dl → 0,
dK (ℓ)
1
= (d − 1) K + sinh  2 K  ln  tanh ( K ) 
dℓ
2
− β K (d , K )
K*= Kc
− β K ( 2, K )
T =∞
K
− β K (1, K )
− β K (1/ 2, K )
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
17
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Migdal-Kadanoff Flow Equations
b = e dl , dl → 0,
dK (ℓ)
1
= (d − 1) K + sinh  2 K  ln  tanh ( K ) 
dℓ
2
− β K (d , K )
K*= Kc
d = 2 : ⇒ sinh [ K c ] = 1
⇒ Kc =
1 
ln 1 + 2 
2
exact!
reason: MK transform.
commutes with duality
transformation!
− β K ( 2, K )
T =∞
K
− β K (1, K )
− β K (1/ 2, K )
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
17
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Migdal-Kadanoff Flow Equations
b = e dl , dl → 0,
dK (ℓ)
1
= (d − 1) K + sinh  2 K  ln  tanh ( K ) 
dℓ
2
− β K (d , K )
K*= Kc
d = 2 : ⇒ sinh [ K c ] = 1
⇒ Kc =
1 
ln 1 + 2 
2
exact!
− β K ( 2, K )
T =∞
reason: MK transform.
commutes with duality
transformation!
1
d = 1 + ε , ε ≪ 1: ⇒ K c ≈
2ε
K
− β K (1, K )
− β K (1/ 2, K )
integrate flow equations:
ν = 1/ yτ ≈ 1/ ε
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
17
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Statistical Landau Theory
(Landau, Ginzburg, Wilson)
• start with microscopic model:
Hmicro [ si ]
• divide system into cells and coarse grain
Z = ∑ exp  −Hmicro [ si ]
{si }


= ∑ ∑ exp  − Hmicro [ si ] ∏ δ  M c , ∑ s j 
j∈c
c
{M c } {si }


= ∑ exp  −Hmeso [ M c ]
aC ≫ a
{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 )
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
18
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Mesoscopic Model
ZH =
∫
configurations [φ ]
D[φ ]exp ( − H[φ ] ) uv cutoff: Λ ∼ 2π / ac
τ
u
2
1

H [φ ] = ∫ d d x  ( ∇φ ) + 0 φ 2 + 0 φ 4 − h φ 
2
4!
2

V
[φ ] = ( d − 2) / 2
µ dimensions:
[τ 0 ] = µ 2
[u0 ] = µ ε
dimensionless interaction constant:
u0 τ 0
−ε / 2
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
19
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Field Theory: Heuristic Considerations
GΛ(2) ( x, T ) ≡ φ ( x + x0 ) φ ( x0 )
(2)
Λ
expect: G ( x, Tc ) ∼ x
− ( d − 2 +η )
G ( x, Tc ) = C x
(2)
Λ
− ( d − 2)
cum
regularized cumulants
but: GΛ(2)  = length − ( d − 2)
( xΛ )
−η
1 + ( xΛ )−ϑ + …


3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
ϑ >0
20
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Field Theory: Heuristic Considerations
GΛ(2) ( x, T ) ≡ φ ( x + x0 ) φ ( x0 )
(2)
Λ
expect: G ( x, Tc ) ∼ x
− ( d − 2 +η )
G ( x, Tc ) = C x
(2)
Λ
− ( d − 2)
cum
regularized cumulants
but: GΛ(2)  = length − ( d − 2)
( xΛ )
−η
1 + ( xΛ )−ϑ + …


ϑ >0
idea: limit Λ → ∞ to extract asymptotic large-x behavior
limit cannot be taken naively!
a) cut-off (to avoid uv divergences)
reason: double role of Λ : 
-1
b)
Λ
= sole length remaining at Tc

3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
20
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Heuristic Intro To Renormalization
trick:
(2)
Λ ,ren
G
( x, Tc , µ ) ≡ ( µ Λ ) GΛ(2) ( x, Tc )
−η
µ : arbitrary momentum scale
(2)
Λ ,ren
G
( x, Tc , µ ) = C x
− ( d − 2)
( xµ )
−η
1 + ( xΛ )−ϑ + …


3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
21
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Heuristic Intro To Renormalization
trick:
(2)
Λ ,ren
G
( x, Tc , µ ) ≡ ( µ Λ ) GΛ(2) ( x, Tc )
−η
µ : arbitrary momentum scale
(2)
Λ ,ren
G
( x, Tc , µ ) = C x
− ( d − 2)
( xµ )
−η
1 + ( xΛ )−ϑ + …


Λ→∞
(2)
Gren
( x, Tc , µ ) = C µ −η x − ( d − 2 +η )
uv finite renormalized function!
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
21
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Heuristic Intro To Renormalization
trick:
(2)
Λ ,ren
G
( x, Tc , µ ) ≡ ( µ Λ ) GΛ(2) ( x, Tc )
−η
µ : arbitrary momentum scale
(2)
Λ ,ren
G
( x, Tc , µ ) = C x
− ( d − 2)
( xµ )
−η
1 + ( xΛ )−ϑ + …


Λ→∞
(2)
Gren
( x, Tc , µ ) = C µ −η x − ( d − 2 +η )
uv finite renormalized function!
G ( x, Tc , µ ) = φ ( x + x0 ) φ
(2)
ren
with
ren
φ ( x ) ≡ Zφ
ren
−1/ 2
φ ( x) ,
ren
( x0 )
cum
Zφ ( µ Λ ) ∼ ( µ Λ )
η
amplitude renormalization
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
21
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
UV Divergences
H=
G
(N )
∫
1
τ 0 2 u0 4 
2
φ +
φ 
d x  (∇φ ) +
 2

2
4!
d
( x1 , … , x N ) ≡ φ ( x1 )⋯φ ( x N )
cum



d 
(N)

ɶ
= FT G (q1 , … , q N ) (2π ) δ  ∑ q j 
 j


-1
ɶ ( 2) (q ) = q 2 + τ −Σ (q )
1 Gɶ (2) (q ) ≡ Γ
0
= q 2 + τ0 +
+
+
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
+…
22
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
UV Divergences
H=
G
(N )
∫
1
τ 0 2 u0 4 
2
φ +
φ 
d x  (∇φ ) +
 2

2
4!
d
( x1 , … , x N ) ≡ φ ( x1 )⋯φ ( x N )
cum



d 
(N)


ɶ
= FT G (q1 , … , q N ) (2π ) δ  ∑ q j 

 j


-1
ɶ ( 2) (q ) = q 2 + τ −Σ (q )
1 Gɶ (2) (q ) ≡ Γ
0
= q 2 + τ0 +
u
=− 0
2
+
+
+…
Λ d − 2 + Cd τ 0 Λ d − 4
ddq
1
∼ 2
2
d
∫
Λ + C4 τ 0 ln Λ , for d = 4
q ≤Λ (2π ) q + τ 0
Λ d − 4
∼
ln Λ , for d = 4
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
22
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
UV Divergences
H=
G
(N )
∫
1
τ 0 2 u0 4 
2
φ +
φ 
d x  (∇φ ) +
 2

2
4!
d
( x1 , … , x N ) ≡ φ ( x1 )⋯φ ( x N )
cum



d 
(N)


ɶ
= FT G (q1 , … , q N ) (2π ) δ  ∑ q j 

 j


-1
ɶ ( 2) (q ) = q 2 + τ −Σ (q )
1 Gɶ (2) (q ) ≡ Γ
0
= q 2 + τ0 +
u
=− 0
2
+
+
q2 ln Λ
divergence
+…
Λ d − 2 + Cd τ 0 Λ d − 4
ddq
1
∼ 2
2
d
∫
Λ + C4 τ 0 ln Λ , for d = 4
q ≤Λ (2π ) q + τ 0
Λ d − 4
∼
ln Λ , for d = 4
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
22
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Renormalization
H=
∫
1
τ 0 2 u0 4 
2
φ +
φ 
d x  (∇φ ) +
 2

2
4!
d
φ ( x ) = Zφ 1/ 2φ ren ( x )
amplitude
τ 0 = τ 0,c + µ 2 Zτ τ
temperature (“mass”)
u0 = µ ε Z u u
coupling constant
uv divergent ∼ ln Λ for d = 4
Zφ , Zτ , Z u = 
uv finite for d < 4
τ 0,c
Λ 2 for d = 4
∼  d −2
for d < 4
Λ
φ 4 theory:
d ≤4
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
23
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Renormalization
H=
∫
1
τ 0 2 u0 4 
2
φ +
φ 
d x  (∇φ ) +
 2

2
4!
d
φ ( x ) = Zφ 1/ 2φ ren ( x )
amplitude
τ 0 = τ 0,c + µ 2 Zτ τ
temperature (“mass”)
u0 = µ ε Z u u
coupling constant
uv divergent ∼ ln Λ for d = 4
Zφ , Zτ , Z u = 
uv finite for d < 4
τ 0,c
Λ 2 for d = 4
∼  d −2
for d < 4
Λ
theorem (Bogoliubov, Parasiuk, Hepp, Zimmermann) for
renormalizable theories:
At any order of perturbation theory all uv singularities
can be absorbed by a finite # of counterterms
(Zφ , Zτ , Z u and τ 0,c ) such that the G
(N)
ren
are uv finite.
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
φ 4 theory:
d ≤4
23
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
RG Equations
bare cumulants:
independent of µ
d
GΛ( N ) ( x ;τ 0 , u0 ) = 0
dµ 0
(N)
Gren
( x ;τ , u , µ , Λ ) =  Zφ ( u , µ Λ ) 
µ ∂µ
−N / 2
GΛ( N )  x ;τ 0 (τ , u , µ , Λ ) , u0 (τ , u , µ , Λ ) 
beta function:
0
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
24
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
RG Equations
bare cumulants:
independent of µ
d
GΛ( N ) ( x ;τ 0 , u0 ) = 0
dµ 0
(N)
Gren
( x ;τ , u , µ , Λ ) =  Zφ ( u , µ Λ ) 
−N / 2
GΛ( N )  x ;τ 0 (τ , u , µ , Λ ) , u0 (τ , u , µ , Λ ) 
N  (N )

RGE:  µ ∂ µ + β u ∂ u + ( 2 + ητ )τ ∂τ + ηφ  Gren ( x ; u ,τ , µ ) = 0
2 

µ ∂µ
beta function:
0
“exponent
functions”:
β u (u, ε ) = µ ∂ µ 0 u
ηφ (u ) = µ ∂ µ 0 ln Zφ
ητ (u ) = µ ∂ µ 0 ln Zτ
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
24
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Scale Invariance at Fixed Points
assumption:
∃ u * such that β u ( ε , u *) = 0
N  (N )

RGE: µ ∂ µ + β u ∂ u + ( 2 + ητ )τ ∂τ + ηφ Gren ( x ; u ,τ , µ ) = 0

2 
(2)
Gren
( x ; u ∗ ,τ , µ ) = µ −η x
−( d − 2 +η )
Ξ (2) ( µ x )τ ν ; u ∗ 
scale invariance for u = u* !
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
25
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Scale Invariance at Fixed Points
assumption:
∃ u * such that β u ( ε , u *) = 0
N  (N )

RGE: µ ∂ µ + β u ∂ u + ( 2 + ητ*)τ ∂τ + ηφ* Gren ( x ; u ,τ , µ ) = 0

2 
(2)
Gren
( x ; u ∗ ,τ , µ ) = µ −η x
−( d − 2 +η )
Ξ (2) ( µ x )τ ν ; u ∗ 
scale invariance for u = u* !
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
25
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Scale Invariance at Fixed Points
∃ u * such that β u ( ε , u *) = 0
assumption:
N  (N )

RGE: µ ∂ µ + β u ∂ u + ( 2 + ητ*)τ ∂τ + ηφ* Gren ( x ; u ,τ , µ ) = 0

2 
(N)
ren
G
( x ; u ,τ , µ ) = µ
∗
dN = − ( d − 2) 2
dN
( µ x)
η = ηφ∗
(2)
Gren
( x ; u ∗ ,τ , µ ) = µ −η x
dN −N η / 2
Ξ ( µ x )τ ν ; u ∗ 
ν = 1 ( 2 + ητ∗ )
−( d − 2 +η )
1/ξ
Ξ (2) ( µ x )τ ν ; u ∗ 
scale invariance for u = u* !
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
25
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Scale Invariance at Fixed Points
∃ u * such that β u ( ε , u *) = 0
assumption:
N  (N )

RGE: µ ∂ µ + β u ∂ u + ( 2 + ητ*)τ ∂τ + ηφ* Gren ( x ; u ,τ , µ ) = 0

2 
(N)
ren
G
( x ; u ,τ , µ ) = µ
∗
dN = − ( d − 2) 2
dN
( µ x)
η = ηφ∗
(2)
Gren
( x ; u ∗ ,τ , µ ) = µ −η x
dN −N η / 2
Ξ ( µ x )τ ν ; u ∗ 
ν = 1 ( 2 + ητ∗ )
−( d − 2 +η )
1/ξ
Ξ (2) ( µ x )τ ν ; u ∗ 
scale invariance for u = u* !
∃ nontrivial fixed points? What if u ≠ u ∗? (generic case)
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
25
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Beta Functions
βu ε
ir-stable
ε ≡ 4−d > 0
u* = O ( ε )
for b → ∞
Gaussian
fixed point
u u*→
d >4
ε =0
d =4
d <4
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
26
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Characteristics
µ → µ (b ) = µ b
d N

 (N)
b
u
−
+
η
(
)
 db 2 φ
 Gren ( x ; u ,τ , µ ) = 0
d
u ( b ) = β u u ( b ) 
db
d
b τ ( b ) = 2 + ητ u ( b )  τ ( b )
db
u ( b = 1) = u
−b
flow equations:
{
}
τ ( b = 1) = τ
ωu = ∂β u ∂u u > 0
u (b) − u ∗ ∼ b −ωu (u − u ∗ )
∗
τ (b) = b yτ Eτ [u , u ]τ ≈ b yτ Eτ [u ∗ , u ]τ
yτ ≡ 1 ν = 2 + ητ
∗
(upon inclusion of h)
nonuniversal
scale factors
h (b) = b yh Eh [u , u ]τ ≈ b yh Eh [u ∗ , u ] h
yh ≡ ∆ ν = ( d + 2 − ηφ∗ ) 2
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
27
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Upshot
(N)
Gren
( x ; u,τ , µ ) = b− Nη 2 EG [u , u ] Gren( N ) ( x ; u ,τ , h , µ / b )
(N)
≈ b − d N − Nη 2 EG u ∗ , u  Gren
( x ; u ∗ ,τ , h , µ )
power of Eh
scaling
function
• universality (crit. expo’s, scaling functions)
• two-scale factor universality
• corrections to scaling from terms ∼ b −ωu ( u − u ∗ )
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
28
H. W. Diehl (Essen): Critical Behavior II: Renormalization Group Theory
Upshot
(N)
Gren
( x ; u,τ , µ ) = b− Nη 2 EG [u , u ] Gren( N ) ( x ; u ,τ , h , µ / b )
(N)
≈ b − d N − Nη 2 EG u ∗ , u  Gren
( x ; u ∗ ,τ , h , µ )
power of Eh
scaling
function
• universality (crit. expo’s, scaling functions)
• two-scale factor universality
• corrections to scaling from terms ∼ b −ωu ( u − u ∗ )
spatial isotropy + short-range interactions + scale invariance
-> conformal invariance! (Polyakov, Belavin, Zamolodchikov, Cardy…)
3rd Summer School on Condensed Matter Research: "Phase Transitions", Zuoz, Engadin (Ch), August 7-14 2004
28