AN INTRODUCTION TO THE FUNCTIONAL ...pdf).pdfEffective Average Action Method Wetterich, Christof. "Exact evolution equation for the effective potential." Physics Letters B 301.1 (1993):

AN INTRODUCTION TO THE FUNCTIONAL RENORMALIZATION GROUP:!

SCALING SOLUTIONS IN CONTINUOUS DIMENSION

RICCARDO BEN ALÌ ZINATI

9 Giugno 2014

[Based on arXiv:1204.3877]

Wilson RG Functional RG ConclusionsOutline

Wilson Renormalization Group

Functional Renormalization Group

OUTLINE OF THE TALK

Effective Average action formalismExact RG equation

Derivative Expansion

Scaling solutions

Critical exponents

Z2 models

Z2 -models universality classes

WILSON’S RENORMALIZATION GROUP

Wilson RG Functional RG ConclusionsOutline Z2 models

Wilson RG Functional RG Conclusions

Close to a second order phase transition

⇠

a ⇠ ⇤�1 lattice spacing

correlation length

Outline Z2 models

Close to a second order phase transition

⇠

a ⇠ ⇤�1 lattice spacing

correlation length

At the critical point :

⇠ % 1 =) ⇠ � a ⇠ ⇤�1


short distance physics is completely washed out!

Z2 models



⇠ % 1 =) ⇠ � a ⇠ ⇤�1




⇠ % 1 =) ⇠ � a ⇠ ⇤�1

WILSON’S IDEA

Built an effective theory for the long distance degrees of freedom




WILSON’S IDEA

⇠ % 1 =) ⇠ � a ⇠ ⇤�1

IMPLEMENTATION

Built an effective theory for the long distance degrees of freedom

RG Transformation = Coarse Graining + Rescaling


Z =X

{�i}

e�H({�i}, ~K) =X

{�B}

X

�i2B

e�H({�i}, ~K) ⌘X

{�B}

e�H({�B}, ~K0)

Kadanoff, Leo P. "Scaling laws for Ising models near Tc." Physics 2.263 (1966): 12.



Z =

Z ⇤

D�(p) e�H[�, ~K] =

Z ⇤/s

D�<

"Z ⇤

⇤/s

D�> e�H[�<,�>, ~K]

#=

Z ⇤

D�0(p0) e�H[�0, ~K0]

Wilson, Kenneth G., and John Kogut. "The renormalization group and the ϵ expansion." Physics Reports 12.2 (1974): 75-199.



!𝓡 ≔ 𝓒 ∘ 𝓢

Features of RG transformation:


!𝓡 ≔ 𝓒 ∘ 𝓢

Probes the system at different scales



Z2 models

!𝓡 ≔ 𝓒 ∘ 𝓢


Preserves the partition function



Z2 models

!𝓡 ≔ 𝓒 ∘ 𝓢


Maps Hamiltonians in Hamiltonians




Z2 models

!𝓡 ≔ 𝓒 ∘ 𝓢


Integrates out short distance degrees of freedom to obtain an effective theory for the long distance ones





Z2 models

!𝓡 ≔ 𝓒 ∘ 𝓢





Generate a flow in the ( - dimensional) parameter space~K ! R( ~K) =: ~K 0

1



Z2 models

!𝓡 ≔ 𝓒 ∘ 𝓢





Generate a flow in the ( - dimensional) parameter space~K ! R( ~K) =: ~K 0

1



Z2 models

Critical Surface:

Mc := { ~K 2 ⌦ | ⇠ = 1}

Fixed Points~K⇤ = R( ~K⇤, s)

~K⇤~Kc

⇠ = 1

⌦physical line

~K ! R( ~K) =: ~K 0


Mc := { ~K 2 ⌦ | ⇠ = 1}


~K⇤~Kc

⇠ = 1

⌦physical line

Hypothesis:

For points in a (finite or infinite) domain on the critical surface, the RG flow converges

to a fixed point


Critical Surface:

Z2 models

~Kc~K⇤

⇠ = 1

⌦Mc := { ~K 2 ⌦ | ⇠ = 1}


physical line

~K0


to a fixed point


Hypothesis:

Critical Surface:

Z2 models

~Kc~K⇤

⇠ = 1

⌦Mc := { ~K 2 ⌦ | ⇠ = 1}


physical line

~K0

UNIVERSALITY

All the theories in the same basin of attraction

belongs to the same universality class


to a fixed point


Hypothesis:

Critical Surface:

Z2 models

FUNCTIONAL RENORMALIZATION GROUP




Main formulations

Wilson-Polchinski Approach

Polchinski, Joseph. "Renormalization and effective Lagrangians."

Nuclear Physics B 231.2 (1984): 269-295.

Effective Average Action Method

Wetterich, Christof. "Exact evolution equation for the effective potential."

Physics Letters B 301.1 (1993): 90-94.

Z2 models


Wilson-Polchinski Approach

Polchinski, Joseph. "Renormalization and effective Lagrangians."

Nuclear Physics B 231.2 (1984): 269-295.

Effective Average Action Method

Wetterich, Christof. "Exact evolution equation for the effective potential."

Physics Letters B 301.1 (1993): 90-94.

Main formulations


Z2 models

WHY FUNCTIONAL RENORMALIZATION GROUP ?




Good convergence properties

Z2 models



Recover known results -expansion, loop expansion,…✏


Z2 models




New approximation schemes: go beyond perturbation theory


Z2 models




Computation valid for any dimension d



Z2 models




Computation valid for any dimension d


Find systematically all possible non-perturbative massless continuum limits for QFTs


Z2 models

Effective Action

Z [J ] =

ZD� e�S[�]+

Rd

dx �(x)J(x)Partition Function


Effective Action

Z [J ] =

ZD� e�S[�]+

Rd


W [J ] = logZ [J ]Free Energy


Effective Action

Z [J ] =

ZD� e�S[�]+

Rd


�W

�J(x)= h�(x)ic =: '(x)W [J ] = logZ [J ]Free Energy


Effective Action

Z [J ] =

ZD� e�S[�]+

Rd


�W


� ['] = �W [J ] +

Zd

dx '(x)J(x)Gibbs Free Energy


Effective Action

Partition Function

�W


� ['] = �W [J ] +

Zd

dx '(x)J(x)Gibbs Free Energy

More transparent physical interpretation

Generates 1PI Graphs


Z [J ] =

ZD� e�S[�]+

Rd

dx �(x)J(x)

Z2 models

Built a one parameter family of functionals interpolating between bare action

and effective actionS

�

Effective Average Action Idea


Built a one parameter family of functionals interpolating between bare action

and effective actionS

�

Effective Average Action Idea


How to do it?

Z2 models

Effective Average Action

e��['] =

ZD� e�S['+�]+

Rd

d

x

��'(x)�(x)


Add the (IR) cutoff term

e��['] =

ZD� e�S['+�]+

Rd

d

x

��'(x)�(x)

e��k

['] :=

ZD� e�S['+�]��S

k

[�]+Rd

d

x

��k

�'(x)�(x)




Z2 models

e��['] =

ZD� e�S['+�]+

Rd

d

x

��'(x)�(x)

e��k

['] :=

ZD� e�S['+�]��S

k

[�]+Rd

d

x

��k

�'(x)�(x)

Γ =S≡ k �k=⇤ ['] = S[� = ']

no fluctuation has been integrated out




Z2 models

e��['] =

ZD� e�S['+�]+

Rd

d

x

��'(x)�(x)

e��k

['] :=

ZD� e�S['+�]��S

k

[�]+Rd

d

x

��k

�'(x)�(x)

�k=0 ['] = � [']

�k=⇤ ['] = S[� = ']

no fluctuation has been integrated out

all fluctuations are integrated out

Γk

k




Z2 models

Universal properties won’t depend on the cutoff chosen!

�Sk [�] =1

2

Zddq

(2⇡)d�(�q)Rk(q)�(q)The (IR) cutoff term

Rmassk (q) = k2

Ropt

k (q) = (k2 � q)✓(k2 � q)

Rexp

k (q) =q

eq/k2 � 1

Rk(q)

q


Take the scale derivative of

EXACT RG EQUATION FOR THE EEA

Wetterich, Christof. "Exact evolution equation for the effective potential." Physics Letters B 301.1 (1993): 90-94.

@t�k ['] =1

2Tr

⇢h�(2)k ['] +Rk

i�1@tRk

�

e��k

['] :=

ZD� e�S['+�]��S

k

[�]+Rd

d

x

��k

�'(x)�(x)



functional integro-differential non-linear exact equation

EXACT RG EQUATION FOR THE EEA

one-loop structure

IR and UV finite

=1

2@t�k['] =

1

2Tr

⇢h�(2)k ['] +Rk

i�1@tRk

�

Z2 models

How to solve it?


How to solve it?

Vertex ExpansionExpand in powers of the field

Iterative solutionPick a seed and solve by iteration


Derivative ExpansionExpand in powers of momenta

Z2 models

How to solve it?

Vertex ExpansionExpand in powers of the field

Iterative solutionPick a seed and solve by iteration




�k['] =

Zddx

Vk(') +

1

2Zk(') (@')

2�+O(@4)

Project the exact FRG equation to obtain a set of coupled p.d.e. involving the running functions:

@tVk(')

@tZk(')

...


Make a truncation Ansatz

�k['] =

Zd

dx

Vk(') +

1

2(@')2

�

Derivative Expansion: LPA

Nicoll, J. F., T. S. Chang, and H. E. Stanley. "Approximate renormalization group based on the Wegner-Houghton differential generator." Physical Review Letters 33.9 (1974): 540.



�k['] =

Zd

dx

Vk(') +

1

2(@')2

�

Calculate the Hessian

�

2�k[']

�'(x)�'(x0)=

⇥�@

2 + V

00k (�)

⇤�(x� x

0)





�k['] =

Zd

dx

Vk(') +

1

2(@')2

�


�

2�k[']

�'(x)�'(x0)=

⇥�@

2 + V

00k (�)

⇤�(x� x

0)

Insert into the ERG equation

@t�k['] = Tr@tRk(�@2)

�@2 + V 00k (') +Rk(�@2)





�k['] =

Zd

dx

Vk(') +

1

2(@')2

�


�

2�k[']

�'(x)�'(x0)=

⇥�@

2 + V

00k (�)

⇤�(x� x

0)

Insert into the ERG equation

@t�k['] = Tr@tRk(�@2)

�@2 + V 00k (') +Rk(�@2)

and project

@tVk(') =1

2(4⇡)d/2�(d/2)

Z 1

0dz zd/2�1 @tRk(z)

�z + V 00k (') +Rk(z)



Choose a cut off shape function to obtain an explicit solution

@tVk(') = cdkd

1 +V 00k (')k2




@tVk(') = cdkd

1 +V 00k (')k2

Introduce dimensionless variables to implement scaling

' = kd/2�1' Vk(') = kdVk(')

to obtain the p.d.e. for the effective dimensionless potential

@tVk(') = �d Vk(') +

✓d

2� 1

◆' V 0

k(') +cd

1 + V 00k (')


S



@tVk(') = cdkd

1 +V 00k (')k2

' = kd/2�1' Vk(') = kdVk(')

to obtain the p.d.e. for the effective dimensionless potential

@tVk(') = �d Vk(') +

✓d

2� 1

◆' V 0

k(') +cd

1 + V 00k (')



Introduce dimensionless variables to implement scaling S

Z2 models

@tVk(') = �d Vk(') +

✓d

2� 1

◆' V 0

k(') +cd

1 + V 00k (')

Scaling Solutions Z2 models LPA


@tVk(') = �d Vk(') +

✓d

2� 1

◆' V 0

k(') +cd

1 + V 00k (')

Ordinary Differential Equation

Scaling solution

@tV⇤(') = 0

�dV⇤(') +d� 2

2'V 0

⇤(') + cd1

1 + V 00⇤ (')

= 0



Z2 models

�dV⇤(') +d� 2

2'V 0

⇤(') + cd1

1 + V 00⇤ (')

= 0

V 0⇤(0) = 0

V⇤(0) =cd/d

1 + V 00⇤ (0)

V 00⇤ (0) =: �

(Initial conditions

Z2 symmetry



�dV⇤(') +d� 2

2'V 0

⇤(') + cd1

1 + V 00⇤ (')

= 0

V 0⇤(0) = 0

V 00⇤ (0) =: �

(Initial conditions

Singularities

'ds(�) := {' | V (') % 1, 8 {d,�}}

Search for universality classes Plot the function ! 'ds(�)



Requiring the potential to be well defined 8' 2 R

Z2 models


'ds(�)

�

Universality classes appear as “spikes”

Z2 models


'ds(�)

�

Ising

Z2 models


'ds(�)

�

Ising

Ising Tri-critical

Z2 models


'ds(�)

�

Ising Tri-critical

Ising Tetra-critical

Ising

Z2 models


'ds(�)

�

'ds(�)

d=2.01 Toward CFT minimal models!

Z2 models

Scaling Solutions Z2 models LPA’

�dV⇤(') +d� 2 + ⌘

2'V 0

⇤(') + cd1� ⌘

d+2

1 + V 00⇤ (')

= 0

⌘k = cd

hV 000k ('0)

i2

h1 + V 00

k ('0)i4

Add the anomalous dimension!

Solve iteratively



0.4

0.1

0.2

0.3

3.02.82.62.42.2

⌘i

d

Anomalous dimension of the first five multi-critical scaling solution as a function of d.

⌘i

Z2 models


Anomalous dimension of the first ten multi critical scaling solution in dimension d=2.

⌘i

@2O( )LPA’

CFT0.4

0.1

0.2

0.3

2 4 6 8 10

⌘i

i

Z2 models

Critical exponents LPA

@tVk(') = �d Vk(') +

✓d

2� 1

◆' V 0

k(') +cd

1 + V 00k (')


@tVk(') = �d Vk(') +

✓d

2� 1

◆' V 0

k(') +cd

1 + V 00k (')

Vk(') = �0 +�2

2!'2 +

�4

4!'4 +

�6

6!'6 + . . .Expand

@tVk @t�2n



Z2 models

@tVk(') = �d Vk(') +

✓d

2� 1

◆' V 0

k(') +cd

1 + V 00k (')

Vk(') = �0 +�2

2!'2 +

�4

4!'4 +

�6

6!'6 + . . .Expand

�2n = @t�2n

All the beta functions for the running couplings can be extracted

�2n =(2n)!

n!

@n

@('2)n@tVk(')

��'=0

The effective potential is the generating function for Beta functions



Z2 models

Mnm =@�2n

@�2m

��⇤

Compute stability matrix

Eigenvalues ⇤1(d) < 0 < ⇤2(d) < ⇤3(d) < . . .

⌫(d) = � 1

⇤1(d)Correlation length critical exponent



Z2 models

↵+ 2� + � = 2

↵+ �� + � = 2

⌫(2� ⌘) = �

↵+ ⌫d = 2

Scaling laws


⌘⌫↵��

�

LPA’ Exact LPA’ World best

0,436 0,25 0,11 0,036

1,05 1 0,65 0,63

-0,11 0 0,06 0,11

0,23 0,125 0,36 0,33

1,65 1,75 1,22 1,24

10,17 15 4,60 4,79

d=2 d=3

Belavin Alexander A., Alexander M. Polyakov, and Alexander B. Zamolodchikov. "Infinite conformal symmetry in two-dimensional quantum field theory." Nuclear Physics B 241.2 (1984): 333-380.

Critical exponents for the Ising universality class in d=2 and in d=3 compared to exact result and world best estimates

Pelissetto Andrea, and Ettore Vicari. "Critical phenomena and renormalization-group theory." Physics Reports 368.6 (2002): 549-727.


Follow, continuously with d, the evolution of RG fixed points through the functional theory space

of effective potentials

CONCLUSIONS




Validation of the correspondence between Landau-Ginsburg actions and minimal models of CFT


CONCLUSIONS

Z2 models




LPA captured qualitatively all possible critical behaviour associated with Z2-simmetry in any d 2.�


CONCLUSIONS

Z2 models




LPA captured qualitatively all possible critical behaviour associated with Z2-simmetry in any d 2.

LPA’ quantitatively reproduced well the known results for critical exponents

�


CONCLUSIONS

Z2 models

THANK YOU


MAIN REFERENCES


Berges Jürgen, Nikolaos Tetradis, and Christof Wetterich. "Non-perturbative renormalization flow in quantum field theory and statistical physics."

Physics Reports 363.4 (2002): 223-386.

Delamotte Bertrand. "An introduction to the nonperturbative renormalization group."

Renormalization Group and Effective Field Theory Approaches to Many-Body Systems. Springer Berlin Heidelberg, 2012. 49-132.

Codello Alessandro. "Scaling solutions in a continuous dimension."

Journal of Physics A: Mathematical and Theoretical 45.46 (2012): 465006.

Morris Tim R. "Elements of the continuous renormalization group."

arXiv hep-th/9802039 (1998).

Mussardo, Giuseppe. Statistical field theory.

Oxford Univ. Press, 2010.

Z2 models

AN INTRODUCTION TO THE FUNCTIONAL ...pdf).pdfEffective Average Action Method Wetterich, Christof. "Exact evolution equation for the effective potential." Physics Letters B 301.1 (1993):

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