sm421 rg

Introduction The Renormalization Group Transformation for the General Case Landau-Ginzburg Hamiltonian

Renormalization Groupclassic paper: K.G.Wilson - RMP, Vol.47,No.4,1975,pp-773-840

Debanjan Basu

Inidian Institute of Science Education and Research

Monday, March 21


Phase Transitions

Phase Transitions

Phase Transitions are a fascinating class of phenomena in naturethat were very di�cult to explain in terms of microscopic quantities

for a long time, even after the advent of statistical mechanics.Moreover the phase transitions exhibit an universality in behaviour

and in critical exponents.


Phase Transitions

Ising Model

Let us de�ne our system - imagine a 1-dimesional lattice ofatoms/molecules. According to the Ising Model, the magneticproperties (i.e. Ferromagnetic and Paramagnetic Properties) of alattice could be suitably modelled by the Hamiltonian:

Hising = ∑〈ij〉

JSzi S

zj −∑

i

Szi h

where the summation over 〈ij〉 indicates summation over nearestneighbours in the lattice only, J being the coupling constantdictating Ferromagnetic (J < 0) and Paramagnetic (J > 0)behaviours respectively. Sz

i obviously denotes the value of the spinmomentum observable at site i .h is the magnetic �eld intensity along z-direction.


Phase Transitions

De�ning the system

This system, as can be seen by inspection would go fromferromagnetic to paramagetic as the J is increased.We now go over to a coarse grained version of this, in thecontinuum limit:De�ne M~k

≡ ∑n exp(i~k ·~x)Szn

Now we can average over a lengthscale of 1/Λ at each point as,

M(~x) =∫ Λ0

ddk(2π)d

exp(−i~k �~x)M~k


Phase Transitions

Partition Function

And then the partition function can be rede�ned,

exp(H [m]) =

[∑{sm}

Λ

∏k=0

δ

(Mk −∑

n

exp(i~k ·~x)Szn

)]exp

(−H0

kT

)

Z =Λ

∏k=0

∫∞

−∞

dMk exp(H [M]) =∫

∞

−∞

DM exp(H [M])

is a functional integral in the continuum limit.


Universal Critical Exponents


The following critical exponents turn out to be universal, i.e. samefor a large class of systems.Critical Exponents of phase transitions are de�ned as follows-

t ≡ T −Tc

Tc

1 Speci�c Heat: C ∼ |t|−α

2 Spontaneous Magnetisation:limH→0+ M ∝ |t|−β

3 Zero Field Susceptibility:(

∂M∂H

)H=0

∼ |t|−γ

4 At T=Tc , M ∼ h1δ

5 Correlation Length, ξ ∼ |t|−ν



The Renormalization Group in a ji�y!A short and forced introduction to Renormalization Group Transformations

De�nition of the Renormalization Group Transformation

exp(H Λ

2[M])

=Λ

∏|k|> Λ

2

∫∞

−∞

dMk exp(HΛ [M])

choose dimensionless momentum parameter q = kΛ , given the cuto�

Λ, i.e. 0< |q|< 1.Accordingly, Mk = zΛσq, where zΛ is chosen such that the average�uctuation amplitude of σq ∼ 1.Note that the σq and σq′ are di�erent in properties therefore,

introduce ς =z Λ2

zΛ, and hence

σq = ςσ2q



RG Transformations

De�nition of the Renormalization Group Transformation

And hence the Renormalization Group Transformation is given by,

exp(H Λ

2

[σ′])=

1

∏q> 1

2

∫∞

−∞

dσq exp(HΛ [σ ])



RG Transformations

An alternate Notation for RG in real space

Coarse Graining: Lengthscale required for writing down theLandau-Ginzburg E�ective Hamiltonian - latticespacing �a�. To decrease the resolution, a→ ba

:(b > 1)

mnewi (x)→ 1

bd

∫b times cell size

ddx ′moldi (x ′)

Rescale: xnew =xold

bRenormalise:

mnewi (x)→ 1

ςbd

∫b times cell size

ddx ′moldi (x ′)


The Renormalization Group Transformation for the GeneralCase

βH =∫ddx

[t2m2 +um4 + vm6 + ...+ K

2(∇m)2 + L

2

(∇2m

)2+ ...

]this already has rotational symmetry and is completelyspeci�ed by a tuple :S = (t,u,v , ....,K ,L, ...)

Apply RG:

coarse grain by b

rescale by x ′ = xb

renomalize as m′ = mς

m′(x ′) =1

ςbd

∫cell of size bcentred at bx ′

ddx m(x)

Given that P [m(x)] ∝ exp (−βH [m(x)]), construct P [m′(x ′)]- this is usually the hardest part!


General RG Transformations - Fixed Point

Find the parameter space point corresponding to the new H .Now this new point describes a Landau Ginzburg Hamiltoniansince the RG procedure respects Rotational Symmetry, andhence,

S ′ = RbS

and this is a map from the parameter space onto itself.

Fixed Point: Lets assume that ∃S∗ � RbS∗ = S∗

Since on application of RG, ξ (S) = bξ (RbS), at the �xedpoint S∗, ξ is either 0 or ∞.Fixed point with ξ = ∞ =⇒ Critical Point at T = TC .


General RG Tranformations - Linearise around �xed point

Linearise in the vicinity of the �xed point:Diagonalize Rbwith the eigenvectors Oi and the eigenvaluesλi (b)The eigenvectors Oi have the interpetation of scaling directionsFrom the physical interpretation of the RG procedure,

RbRb′Oi = λi (b)λi (b′)Oi =⇒ Rbb′Oi = λi (bb

′)Oi

Rb being the dilation operator. Along with λi (1) = 1, thisimplies that λi (b) = byi , where yi are called anomalousdimensions.


General RG Tranformations - Linearise around �xed point(contd.)

In the vicinity of the �xed point -

S = S∗+∑i

giOi

S′= S∗+∑

i

gibyiOi

If yi > 0, gi increases =⇒ Oi is a Relevent Operator

If yi < 0, gi decreases =⇒ Oi is an Irrelevent Operator

If yi = 0, gi is constant =⇒ Oi is a Marginal Operator, which

means that higher order terms are required to predict the

behaviour in this direction

The subspace spanned by the irrelevant operator forms the

basin of attraction.


General RG Tranformations - The Homogeneity Assumption

Homogeneity Assumption:

ξ (g1,g2, ...) = bξ (by1g1, ...)

Enforcing the Homogeneity Assumption about the singularpart of the correlation length to be

ξ (g1,g2, ...) = |g1|−νξ

(1,

g2

|g1|∆,

g3

|g1|∆,

g4

|g1|∆, ...

)Here ∆ is called the Gap Exponent. Obviously, this imposes

by1g1 = 1 =⇒ b = g− 1

y11

and thus,

ξ (g1,g2, ...) = |g1|−1y1 ξ

(1,

g2

|g1|∆,

g3

|g1|∆,

g4

|g1|∆, ...

)and the gap exponents, ∆α = yα

y1.


Landau Ginzburg Hamiltonian

βH = βF0+∫ddx

[t

2m2(x) +um4(x) +

K

2(∇m)2 +

L

2

(∇2m)2

+ . . .

]Built of pieces that satisfy the following conditions:

Uniformity and Locality

Analyticity (not microscopic or macroscopic)

Symmetries of underlying space : Rotation, Translation etc


Solution for Gaussian Model using Renormalization Group

Gaussian Model

We introduce the Gaussian model,

Z =∫

DM exp(βH [M])

=∫

DM exp

(∫ddx

[t

2m2(x) +

K

2(∇m)2 +

L

2

(∇2m)2

+ . . .

])where the um4(x) term is absent from the Landau GinzbergE�ective Hamiltonian.De�ning,

~m(q) =∫ddx exp(iq · x)~m(x)

and

~m(x) =∫

ddq

(2π)dVexp(−iq · x)~m(q)



The Partition Function

In terms of the Fourier Modes,∫ddx [~m(x)]2 =

∫ddx

∫ ∫ddq1d

dq2e−i(q1+q2)x

Vm(q1)m(q2)

=∫ddq

m(q)m(−q)

V

Therefore the Hamiltonian becomes,

βH = ∑q

(t +Kq2 +Lq4 + ...

V

)|m(q)|2−h ·m(q = 0)

Z = ∏q

V−n2

∫dm(q)exp

[− t +Kq2 +Lq4 + ...

V|m(q)|2 +h ·m(q = 0)

]

∴Z ≈∫

Dmexp

{−∫

ddq

(2π)d− t +Kq2 +Lq4 + ...

V|m(q)|2 +h ·m(q = 0)

}



Apply RG

Applying the Renormalization Group Procedure now,

Coarse Grain: average out lengthscale a < x < ba ≡average ofthe wavenumber interval Λ

b< k < Λ. Now we break up the

momenta into two subsets: {m(q)}= {σ(q>)}⊕{m(q<)}

Z =∫

Dm(q<)∫

Dσ(q<)exp(−βH [m,σ ])

= exp

[−n2V

∫ Λ

Λb

ddq

(2π)dln(t +Kq2 +Lq4 + ...

)]×

∫ Λb

0

Dm(q<)exp

{−∫

ddq

(2π)d− t +Kq2 +Lq4 + ...

V|m(q)|2 +h ·m(q = 0)

}



Apply RG

Coarse Grain

Rescale:x ′ =

x

b

q′ = bq

Z = exp(−V δ f

b(t))∫

Dm(q<)exp

{−∫ Λ

b

0

dd q′

(2πb)d− t+Kb−2q′2 +Lb−4q′4 + ...

V|m(q)|2 +h ·m(q = 0)

}



Apply RG

Coarse Grain

Rescale

Renormalize: Either m′(x ′) = 1

ςm(x ′), in the real space, or

m′(q′) = 1

zm(q′) in terms of the fourier modes.

Choice:m′(x ′) = 1

ςm(x ′)

Z = exp(−V δ f

b(t))∫

Dm(q<)exp

{−∫ Λ

b

0

dd q′

(2πb)dz2− t+Kb−2q′2 +Lb−4q′4 + ...

V|m(q)|2 +zh ·m(q = 0)

}



E�ect of RG Transformation in Parameter Space

This is another Gaussian Hamiltonian, with renormalizedparameters,

t ′ = z2b−d t

h′ = zh

K ′ = z2b−d−2K

L′ = z2b−d−4L

. . . . . .



Getting Results

The singular point t=h=0, is mapped onto itself. The choice

z = b1+ d2 sets K ′ = K and makes L′ scale to zero.

Away from criticality, the relevant directions thus scale as -{t ′ = b2t

h′ = b1+ d2

=⇒

{yt = 2

yh = 1+ d2

Now we can move on to identify the critical exponents as

ν =1

yt= 1/2

∆ = ytyh

=1+ d

2

2=

1

2+d

4

α = 2−dν = 2− d

2

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