Renormalization group:

-theory and -expansion

Henri Orland IPhT, CEA-Saclay (France)

CSRC, Beijing (China)

'4 "

Outline• Ising model: from lattice to continuous field theory

• Dimensional analysis, power counting, UV cut-off

• Critical dimension, relevant-irrelevant operators

• Renormalization group: the Gaussian model

• Renormalization group: cumulant expansion

• The fixed points: Gaussian and non-trivial

• -expansion, order one and higher orders.

• Relation to Renormalization theory in QFT

"

Ising model: From lattice model to continuous Field Theory

• Ising model: magnetic spins along the z-direction, in a magnetic field

• Short range interactions between any pair of spins i and j . We study a ferromagnetic system (e.g. nearest neighbor interactions)

• Partition function

Jij = J(ri � rj)

Jij < 0

Si = ±1

Z =X

Si=±1

e�2

Pi,j KijSiSj+�H

Pi Si

Kij = �Jij > 0

• Gaussian integral

• so

• Partition function

ZNY

i=1

dx

i

e

� 12xiAijxj+uixi =

(2⇡)N/2

pdetA

e

12uiA

�1ij uj

e�2 KijSiSj ⇠

Z NY

i=1

d�ie� �

2 �iK�1ij �j+��iSi

Z =X

Si=±1

Z NY

i=1

d�ie� �

2 �iK�1ij �je�

Pi(H+�i)Si

=

Z NY

i=1

d�ie� �

2 �iK�1ij �j+

Pi log 2 cosh �(H+�i)

K(k) =

ZddreikrK(r)

K�1(k) =1

K(k)

Large distance behavior k ! 0

K(k) ⇠k!0 K0(1� ck2) c > 0

K(k)�1 ⇠k!0 K�10 (1 + ck2)

r � a

macroscopic distance

microscopic scale

X

i,j

�iK�1ij �j ⇠ K�1

0

Zddk

(2⇡)d�̃(k)(1 + ck2 + . . .)�̃(�k)

The continuous limit

• If we are interested in correlation functions at distances large compared to lattice spacing ( near a critical point), we can take a continuous limit.

• Exercise: shift field by magnetic field, redefine coefficients and rescale fields, expand log cosh, and show that the partition function can be written as

Z =

ZD�(r)e

�Rddr

12�(r)(�r2+r0)�(r)+h�(r)+

P1m=2 u2m�2m(r)

!

proportional to in mean-fieldT � Tc

Dimensional analysis and power counting: canonical dimensions

• All the analysis and integrals will be done in momentum space (k-space) so do dimensional analysis in momentum space UV cut-off

• It follows that

⇤ =2⇡

aZddr(r�)2 = 1 = ⇤�d+2⇤2[�] ) [�] =

d� 2

2

[r0] = 2

[u0] = [u4] = 4� d

[u2m] = d� (d� 2)m

[h] =d+ 2

2> 0

• As we will see, an operator is relevant, marginal, irrelevant if its dimension is

• The magnetic field is relevant in all dimensions.

• The “mass” is always relevant

• In d=2, all operators are relevant

• In d=3, only relevant, marginal

• In d>3, only relevant

� > 0, � = 0, � < 0�

�4 �6

�4

r0

The field-theory for critical phenomena

• As we will discuss a specific scheme of calculation (epsilon expansion), we will work around dimension 4. Then only one relevant operator. Near criticality, near dimension 4, the system is thus well described by

• Note that is the difference between terms from

• and from the log cosh, so it can change sign

�4

�4 field-theoryr0

K(k)

Z =

ZD�(r)e

�Rddr

⇣12�(r)(�r2+r0)�(r)+

u04! �

4(r)+h�(r)

⌘

Renormalization group: the Gaussian model

• Near criticality, correlation length

• System is (nearly) scale invariant

• Equivalent of block spins: momentum shell integration

• Originally, UV cut-off

• Define

• (Try to) integrate over the fields

• Rescale momenta from

⇠ � a

⇤ =2⇡

a⇤0 = ⇤� d⇤

⇤0 < k < ⇤

⇤0 to ⇤

The Gaussian ModelAssume u0 = 0

Z =

ZD�(r)e�

Rddr( 1

2�(r)(�r2+r0)�(r)+h�(r))

�<(k) = �(k) i↵ 0 < k < ⇤0

�>(k) = �(k) i↵ ⇤0 < k < ⇤

�(k) = �<(k) + �>(k)

�<(k)�>(k) = 0

, 0 otherwise, 0 otherwise

Z =

ZD�(k) exp

�Z ⇤

0

ddk

(2⇡)d

✓1

2

�(k)(k2 + r0)�(�k)

◆+ h�(k = 0)

!

Note: the model does not have a low T phase

constantRescaling of k q =

⇤

⇤0 k

Rescaling of so that coefficient is 1/2:�< � =

✓⇤0

⇤

◆ d+22

�<

Z = Z1

ZD�<(k) exp

�Z ⇤0

0

ddk

(2⇡)d

✓1

2

�<(k)(k2+ r0)�<(�k)

◆+ h�<(k = 0)

!

Z1 =

ZD�>(k) exp

�Z ⇤

⇤0

ddk

(2⇡)d

✓1

2

�>(k)(k2+ r0)�>(�k)

◆!

Z ⇤0

0ddkk2�<(k)�<(�k) =

✓⇤0

⇤

◆d+2 Z ⇤

0ddq q2�<(q)�<(�q)

⇤0

⇤= 1� d⇤

⇤= 1� dl

System equivalent to original one, for large distancesr00 = (1� dl)�2r0

h0 = (1� dl)�d+22 h

Differential form = Flow equations

dr

dl= 2r

dh

dl=

d+ 2

2h

Canonical dimensions of operators

Relevant operators. Fixed pointsr⇤ = 0

h⇤ = 0Compute exponents!

Z = Z1

ZD�(k) exp

�Z ⇤

0

ddk

(2⇡)d

1

2

�(k)⇣k2 +

✓⇤

0

⇤

◆�2

r0⌘�(�k)

!+

✓⇤

0

⇤

◆� d+22

h�(k = 0)

!

Cumulant expansionInclude the interaction term

Z =

ZD�(r)e

�Rddr

12�(r)(�r2+r0)�(r)+

u04! �

4(r)+h�(r)

!

Take h=0In Fourier space

Z =

ZD�(k) exp

�Z ⇤

0

ddk

(2⇡)d

✓1

2

�(k)(k2 + r0)�(�k)

◆

+

u0

4!

(2⇡)dZ ⇤

0

4Y

i=1

ddki(2⇡)d

�(d)(k1 + k2 + k3 + k4)�(k1)�(k2)�(k3)�(k4)

!

Make separation� = �< + �>

In the high temperature phase (no symmetry breaking), do a cumulant expansion (equivalent to a loop expansion) to integrate over �>

Z(�<) =

ZD�> exp

✓�Z

ddr(H0(�>) + V (�<,�>))

◆

= Z0hexp✓�Z

ddrV (�<,�>)

◆i0

hW (�>)i0 =

1

Z0

ZD�> exp

✓�Z

ddrH0(�>)

◆W (�>)

Z0 =

ZD�> exp

✓�Z

ddrH0(�>)

◆

Z =

ZD�<(r) exp

✓�Z

ddr⇣1

2

�<(r)(�r2+ r0)�<(r) +

u0

4!

�4<(r)

⌘◆

⇥Z

D�>(r) exp

✓�Z

ddr⇣1

2

�>(r)(�r2+ r0)�>(r) +

u0

4!

(4�3<�> + 6�2

<�2> + 4�<�

3> + �4

>)

⌘◆

1 2 3 4

Expanding to the second cumulant

Z = Z0 exp

�Z

ddrhV (�>(r))i0 +1

2

Zddrddr0

⇣hV (�>(r))V (�>(r

0))i0 � hV (�>(r))i0hV>(�(r

0))i0

⌘!

ZddrV (�>) =

u0

4!

Zddr

�4�3

<�> + 6�2<�

2> + 4�<�

3> + �4

>

�

=u0

4!

Z ⇤

⇤0

4Y

i=1

dki(2⇡)d

(2⇡)d�(d)(k1 + k2 + k3 + k4)c↵�↵(k1)�↵(k2)�↵(k3)�↵(k4)

ZddrH0(�>) =

1

2

Zddr �>(r)(�r2 + r0)�>(r)

=1

2

Z ⇤

⇤0

ddk

(2⇡)d�>(k)(k

2 + r)�>(�k)

coeff. 1,4,6

1 2 3 4

Wick TheoremZ

NY

i=1

dx

i

e

� 12xiAijxj+uixi =

(2⇡)N/2

pdetA

e

12uiA

�1ij uj

By taking derivatives w.r.t and then setting ui ui = 0

hxixji = A

�1ij

hxi1xi2 . . . ximi = 0 if m odd

hxi1xi2 . . . ximi =X

all complete sets of pairings

A

�1

i1iP1A

�1

i2iP2. . . A

�1

iniPn

if m evenExample:

Z +1

�1

dxp2⇡

e

� x

2

2

hx2i = 1 Wick hx2ni = (2n� 1)!! = (2n� 1).(2n� 3) . . . 3.1

Cumulant expansionh�>(k)�>(k

0)i = �(d)(k � k0)

k2 + r0

Use Wick theoremOrder 1 in V: only term 2 contributes to r0

Order 2 in V: only term 2-2 contributes to u0

h�2>(r)i =

Z ⇤

⇤0

ddk

(2⇡)d1

k2 + r0

⇡ 2d⇤d�1

(2⇡)dd⇤

⇤2 + r0= Sd⇤d�1

(2⇡)dd⇤

⇤2 + r0

where Sd =2⇡d/2

�(d/2)

1

2

⇣u0

4!

⌘2⇥ 36

Zddrddr0�2

<(r)�2<(r

0)�h�2

>(r)�2>(r

0)i � h�2>(r)ih�2

>(r0)i

�

1

2

⇣u0

4!

⌘2⇥ 2⇥ 36

Zddrddr0�2

<(r)�2<(r

0)h�>(r)�>(r0)i2

g(r � r0) = h�>(r)�>(r0)i =

Z ⇤

⇤0

ddk

(2⇡)deik(r�r0)

k2 + r0

Propagator g is short ranged (because large k): expansion in gradients

g2(r) = �0�(d)(r) + . . .

�0 =

Zddrg2(r)

=

Z ⇤

⇤0

ddk

(2⇡)d1

(k2 + r0)2

= Sd⇤d�1d⇤

(2⇡)d1

(⇤2 + r0)2

Remains to do: 1) rescale momenta from to 2) rescale so that coefficient equal 1/2

⇤0 ⇤�<

dr

dl= 2r +

u

2

Sd

(2⇡)d⇤d

(⇤2 + r)

du

dl= "u� 3

2u2 Sd

(2⇡)d⇤d

(⇤2 + r)2

" = 4� d

dh

dl=

2 + d

2hNote that

r00 = r0 +u0

2Sd

⇤d�1d⇤

(2⇡)d(⇤2 + r0)

u00 = u0 �

3

2u20Sd

⇤d�1d⇤

(2⇡)d(⇤2 + r0)2

Z = C

ZD�<(r) exp

✓�Z

⇤0ddr

⇣1

2

�<(r)(�r2+ r00)�<(r) +

u

4!

�4<(r)

⌘◆

Renormalization Group Equations for running coupling constants (r(l), u(l))

Fixed points: The epsilon expansion

Gaussian: r⇤ = u⇤ = h⇤ = 0

Non trivial: h⇤= 0, r⇤ and u⇤

of order "

To order 1 in " r⇤ = �⇤2

6"

u⇤ =16⇡2

3"

Linearize equations around non trivial fixed point:

y1 = 2� "

3> 0

y2 = �" < 0

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relevant

irrelevant

Critical exponents to order one in epsilon

↵ ="

6

� =1

2� "

6

� = 1 +"

6� = 3 + "

⌘ = 0

" = 4� d

• Relation to Renormalization theory in QFT