The Functional Renormalization Group...The Functional Renormalization Group Andrea Coser SISSA -...

The Functional Renormalization Group

Andrea Coser

SISSA - ISAS

Introduction to the Functional RG

14/09/2012

Outline Universality Wilson RG The EAA method A computation

Outline

1 Critical phenomena and universalityCritical exponentsThe scaling hypothesis

2 The Wilson approach to RenormalizationThe RG flowThe RG flow: around the Fixed Point

3 Functional RG: the EAA methodThe Effective Average ActionThe cutoff functionThe exact RG equation

4 Computation of critical exponents in the O(N) model

The Functional Renormalization Group SISSA-ISAS 2 / 22


Critical phenomena and universality

Phase transitions:

• magnetic system, liquid-gas transition

• non-analyticity of the thermodynamicpotential

• first or second order

P

T

Pc

Tc

solid

gas

liquidC

Second order phase transition −→ different symmetries in the two phases−→ order parameter−→ critical phenomena

Universality:

some properties near the critical point appear to be the same for very differentphysical systems

the behavior is rather independent of the microscopical details (interactionsbetween particles)



Critical exponents

Critical Exponents:

Definitions: near the critical point (t = (T − Tc)/Tc)

• specific heat: C ∼ |t|−α

• order parameter (B = 0): m ∼ |t|β

• susceptibility: χ ∼ |t|−γ

• order parameter (t = 0): m ∼ |h|1δ

• correlation function: G(r , t) =G±(r/ξ(t))

rd−2+η

• correlation length: ξ ∼ |t|−ν

Scaling relations (experimentally verified):

γ = ν (2− η) Fisher

β =1

2(2− α− γ) Rushbrooke

γ = β (δ − 1) Widom

α = 2− νd Josephson

0.10.005 0.01

reduced temperature

0.02 0.05 0.1

0.2

0.3

0.4

0.5

0.6

ma

gn

eti

zati

on

15

Figure: magnetization in nickelit is found β = 0.358



Critical exponents

Example: gas-liquid coexistence region

−→ critical points are very different

−→ reduced quantities lie on the same curve: universality → β = 1/3

00.55

0.60

0.65

0.70

0.75

0.80

0.85

0.90

0.95

1.00

0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6

ρ/ρc

T/T

c

Ne

A

Kr

Xe

N2

O2

CO

CH4

13

Figure: Guggenheim, J. Chem. Phys. 13, 1945

Tc (C) Pc (atm)

Ne -228.7 26.9Ar 1122.3 48Kr -63.8 54.3Xe 16.6 58N2 -147 33.5O2 -118.4 50.1CO -140 34.5CH4 -82.1 45.8



The scaling hypothesis

Scaling hypothesis: at the critical point the correlation lengthis the only characteristic length of the system

Consequences: free energy: f (t, h) ∼t∼0h∼0

|t|2−αF±(

h

|t|∆

)

1 similar scaling laws for the otherquantities

2 scaling relations among exponents

m(t, h) = |t|2−α−∆M±(h/|t|∆

) β = 2− α−∆

∆ = βδ

χ(t, h) = |t|2−α−2∆χ±(h/|t|∆

)−γ = 2− α− 2∆

f (t, 0) ∼ ξ−d ∼ |t|νd νd = 2− α

χ =

∫dd r G(r , t) γ = ν(2− η)

−0.2 0 0.2

80

Cp (

J/m

ole

K)

100

120

T − Tλ (µK)

68

Figure: specific heat at the superfluidtransition in 4He



The Wilson approach to Renormalization

Critical point ξ →∞:

• fluctuations on all wavelengths

• ξ a ∼ Λ−1: short distance details are washed out

Idea: build an effective theory for the long distance degrees of freedom

Implementation: Coarse graining + rescaling −→ RG transformation









Z =∑σi

e−H(σi,~K) =∑σB

∑σi∈B

e−H(σi,~K) ≡∑σB

e−H(σB,~K ′)









Z =

∫ Λ

Dφ(p) e−H[φ,~K ] =

∫ Λ/s

Dφ<

[∫ Λ

Λ/sDφ> e−H[φ<,φ>,~K ]

]=

∫ Λ

Dφ′(p′) e−H[φ′,~K ′]



Coarse graining + rescaling −→ RG transformation

Effects of RG transformation:

• probe the system at different scales

• flow in parameter space ~K → ~K ′ = ~T(~K , s

)(∞-dimensional)

−→ flow in the space of theories

• dimensionless correlation length: ξ =ξ

a

a→ s a ⇒ ξ′ =ξ

s

ξ when s : the coarse grained system is less critical

ξ ∼ |T − Tc |−ν −→ t increases

• ξ =∞ ⇒ ξ′ =∞ −→ the system is still critical

critical surface: set of points with ξ =∞ −→ co-dimension 1



The RG flow

Fixed point:

~K∗ = ~T(~K∗, s

)

Hypothesis:~Kc in a domain of the critical surface:the flow converges to the fixed point

→ all the theories in the basin of attraction belong to the same universality class

~K0 near the critical surface:the flow get close to ~K∗ and then leave

−→ universality



The RG flow: around the Fixed Point

Infinitesimal RG transformation −→ beta function∂~Ks

∂ ln s=∂~T

∂s

∣∣∣∣~Ks ,1

≡ ~β(~Ks)

Linearization around the FP: δ~Ks = ~Ks − ~K∗

sd~Ks

ds=

d~β

ds

∣∣∣∣~K∗︸︷︷︸

stabilitymatrixMij

·δ~Ks +O(δ~K 2

s

)

• scaling directions: eigenvectors of M −→ M~φi = λi~φi

• scaling fields: projection of δ~Ks on ~φi vi (s) =∑α

δKs α φαi

→ not mixed by RG transformations: sdvids

(s) = λivi (s) ⇒ vi (s) ∝ sλi

(a) λi > 0: relevant coupling −→ temperature

(b) λi < 0: irrelevant coupling

(c) λi = 0: marginal coupling



The RG flow: around the Fixed Point

RG transformation for the free energy: f (~K)→ s−d f (~K ′)

f (t, h, v2, . . . ) = s−d f (sλt t, sλhh, sλ3v2, . . . )

choosing s = |t|−1/λt we get

scaling relations:

f (t, h) = |t|dλt f

(±1,

h

|t|λh/λt, 0, . . .

)= |t|

dλt F±

(h

|t|λh/λt, 0, . . .

)

we can compute

critical exponents:

2− α =d

λt∆ =

λh

λt



Functional RG: the EAA method

The Effective Action formalism:

• partition function −→ generator of correlation functions

Z [J] =⟨eJφ⟩

=

∫Dφ e−H[φ]+

∫x J(x)φ(x) 〈φ(x1)φ(x2)〉 =

δ2Z [J]

δJ(x1)δJ(x2)

∣∣∣∣J=0

• connected correlation generating function

W [J] = log⟨eJφ⟩

〈φ(x1)φ(x2)〉c =δ2W [J]

δJ(x1)δJ(x2)

∣∣∣∣J=0

• Effective action (Gibbs free energy)

Legendre transform:ϕJ ≡ 〈φ〉J ⇒ Jϕ = J[ϕ]

Γ[ϕ] = Jϕϕ−W [Jϕ]

(i) function of ϕ, which has a physical interpretation

(ii) generates 1PI graphs

(iii)

(δΓ

δϕδϕ

)−1

x1 x2

=δW

δJ(x1)δJ(x2)



The Effective Average Action

Effective Average Action Γk[ϕ]:

Gibbs free energy of the rapid modes that have been integrated out

k ≤ p ≤ ΛΛ : UV cutoff of the original theory

k : IR cutoff → running parameter

RG flow−−−−→

One parameter family of functionals:

1 k = Λ ⇒ original Hamiltonian: Γk=Λ[ϕ] = H[ϕ]

2 k = 0 ⇒ full Gibbs free energy: Γk=0[ϕ] = Γ[ϕ]



The cutoff function

Implementation:

decouple the slow mode giving them a large mass→ cutoff function Rk(q2)

Zk [J] =

∫Dφ exp

(−H[φ]−∆Hk [φ] +

∫Jφ

)∆Hk [φ] =

1

2

∫q

Rk(q2)φqφ−q

(a) k = 0 → Rk=0(q2) = 0 ∀q

(b) k = Λ → Rk=Λ(q2) = Λ2 ∀q

(c) 0 < k < Λ → Rk(q2 > k2) ' 0

Rk(q2 < k2) ' k2

k2

q2

k2

Rk(q2)

k2

q2

______

eq

2/k

2

-1(k

2-q

2)θ(k

2-q

2)

Cutoff function Rk(q2)

−→ Γk [ϕ] + Wk [J] =

∫Jϕ−

1

2

∫qRk (q2)ϕqϕ−q



The exact RG equation

Exact RG equation for the EAA:

∂kΓk [ϕ] =1

2

∫q

∂kRk(q2)(

Γ(2)k [ϕ] +Rk

)−1

q,−q

Rk (q, q′) =

(2π)dδ(q + q′)Rk (q2)





∂kΓk [ϕ] =1

2

∫q

∂kRk(q2)(

Γ(2)k [ϕ] +Rk

)−1

q,−q

Rk (q, q′) =

(2π)dδ(q + q′)Rk (q2)

Properties:

• integral equation

• differential equation

• non-linear equation

• IR finite

• UV finite





∂kΓk [ϕ] =1

2

∫q

∂kRk(q2)(

Γ(2)k [ϕ] +Rk

)−1

q,−q

Rk (q, q′) =

(2π)dδ(q + q′)Rk (q2)

Approximation schemes:• vertex expansion:

Γk [ϕ] =∞∑n=0

1

n!

∫x1,...,xn

Γ(n)k x1,...,xn

[ϕ] (ϕ− ϕ)x1. . . (ϕ− ϕ)xn

tower of coupled functional equations

with some Ansatz for Γk (truncation) → tower of beta-functions

• derivative expansion:

Γk [ϕ] =

∫x

[1

2Zk (ϕ) (∂ϕ)2 + Uk (ϕ) + higher derivative terms

]effective potential: Uk(ϕunif) ≡

1

ΩΓk [ϕ = ϕunif]



Computation of critical exponents in the O(N) model

Example: O(N)-model

Ansatz for the EAA:

Γk [~ϕ] =

∫x

[1

2∂µϕa∂

µϕa +m2

k

2ϕaϕa +

λk

4!(ϕaϕa)2

], a = 1, . . . ,N

• Local Potential Approximation (LPA)

• Zk = 1 =⇒ ηk ≡ −k∂k lnZk = 0

• truncation of Uk(ϕ) up to ϕ4 : Uk(ϕ) = m2kρ+

λk

3!ρ2, ρ =

1

2ϕaϕa

Flow equation for the Effective Potential:

∂tUk(ρ) =1

2

∫q

∂tRk(q)

(1

q2 + Rk(q) + U ′k + 2ρU ′′k+

N − 1

q2 + Rk(q) + U ′k

)m2

k = U ′k(ρ)|ρ=0 λk = 3U ′′k (ρ)|ρ=0



dimensionless variables: m2k = m2

kk−2 and λk = λkk

d−4

y ≡ q2

k2Rk(q2) ≡ k2 y r(y)

Flow equations:

∂tm2k = −2m2

k +λk

3

N + 2

(4π)dΓ(d2

) ∫ ∞0

dy yd2

+1 r ′(y)

[y(1 + r(y)) + m2k ]

2

∂t λ = (d − 4)λk −2

3λ2k

N + 8

(4π)dΓ(d2

) ∫ ∞0

dy yd2

+1 r ′(y)

[y(1 + r(y)) + m2k ]

3

Choose a cutoff function:

Rk(z) = (k2 − z)θ(k2 − z) , ⇒ r(y) =1− y

yθ(1− y)



∂tm2k = βm2 (m2

k , λk)

= −2m2k −

N + 2

3cd

λk

[1 + m2k ]

2

∂t λk = βλ(m2k , λk)

= (d − 4)λk +2

3(N + 8)cd

λ2k

[1 + m2k ]

3

→ beta functions are non-perturbative

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0

-0.5

0.0

0.5

1.0

1.5

2.0

m k2

Λ

k



∂tm2k = βm2 (m2

k , λk)

= −2m2k −

N + 2

3cd

λk

[1 + m2k ]

2

∂t λk = βλ(m2k , λk)

= (d − 4)λk +2

3(N + 8)cd

λ2k

[1 + m2k ]

3

→ beta functions are non-perturbative

G

WF

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0

-0.5

0.0

0.5

1.0

1.5

2.0

m k2

Λ

k

Fixed points:

m2∗ = − (4− d)(N + 2)

8(N + 5)− d(N + 2)

λ∗ =96

cd

(4− d)(N + 8)2

[8(N + 5)− d(N + 2)]3

• d < 4: two Fixed Points

• d = 4: only the Gaussian FP

• d > 4: spurious FP



Stability matrix:

Mij =

(∂m2βm2 ∂λβm2

∂m2βλ ∂λβλ

)∣∣∣∣∗

Eigenvalues:

Λ± =1

2

(TrM±

√(TrM)2 − 4 detM

)ΛWF± = − 1

N + 8

3(N + 4)− d(N + 5)

±√

3

2d2 (N2 + 10N + 22)− 6d (2N2 + 21N + 52) + (5N + 28)2

Exponent ν:

ν = − 1

ΛWF−

'd→4−

1

2+

N + 2

4(N + 8)(4− d) +O((4− d)2)

• same result of of theε-expansion (at first order)

• the result is valid for any d



Ising (N = 1) massless case (m2k = 0): λk =

(k

Λ

)d−4 λΛ

1 +6ud λΛ

4− d

[(k

Λ

)d−4

− 1

]

d=3:

λk =Λ

k

λΛ

1 + 6u3λΛ

(Λk − 1

) −−−−→k→0

Λ fixed

1

6u3

• k → 0: IR fixed point

• Λ→∞: asymptotic freedom

Π2

0.0 0.5 1.0 1.5 2.0

k

L

2

4

6

8

10

Λ

k

d=4:

λk =λΛ

1− 6u4λΛ log kΛ

−−−−→k→0

Λ fixed

0

• k → 0: triviality

• Λ→∞: Landau pole

1 2 3 4 5

k

L

- 5

0

5

Λ

k



Ising (N = 1) massless case (m2k = 0): λk =

(k

Λ

)d−4 λΛ

1 +6ud λΛ

4− d

[(k

Λ

)d−4

− 1

]d=3:

λk =Λ

k

λΛ

1 + 6u3λΛ

(Λk − 1

) −−−−→k→0

Λ fixed

1

6u3

• k → 0: IR fixed point

• Λ→∞: asymptotic freedom

Π2

0.0 0.5 1.0 1.5 2.0

k

L

2

4

6

8

10

Λ

k

d=4:

λk =λΛ

1− 6u4λΛ log kΛ

−−−−→k→0

Λ fixed

0

• k → 0: triviality

• Λ→∞: Landau pole

1 2 3 4 5

k

L

- 5

0

5

Λ

k


Conclusions References

Conclusions

Summary

• RG explains universality −→ flow to a FP

• RG explains critical behavior −→ critical exponents

• EAA: computational framework −→ approximation schemes

• it is possible to compute universal quantities −→ exponents ν, η

Why Functional RG?

• clearer picture (with respect to perturbative RG)

• recover known results (ε-expansion, loop expansion, . . . )

• new approximation schemes: possibility to go beyond perturbation theory

• computation valid for any dimension d


Conclusions References

References

K. Huang, “Statistical Mechanics”, 1987, Wiley

I. Herbut, “A modern approach to Critical Phenomena”, 2007, CambridgeUniversity Press

B. Delamotte, “An introduction to the nonperturbative renormalizationgroup”, 2007, arXiv:cond-mat/0702365v1


The Functional Renormalization Group...The Functional Renormalization Group Andrea Coser SISSA -...

Documents

Transcript of The Functional Renormalization Group...The Functional Renormalization Group Andrea Coser SISSA -...