Chap. 6 Ginzburg-Landau model and

Landau theory

Youjin Deng09.12.12

From Ising model to the Ginzburg-Landau model

Ising modelUniaxial ferromagnetOne spin variable in each cubic cellEnergy of these spins---Cell Hamiltonian

This is the simplest form to see energy is smaller if the spin agrees with its neighbors

cσ

2'^)(

21][ ∑∑ +−=

cr rccJH σσσ

From Ising model to the Ginzburg-Landau model

Generalize the Hamiltonian slightly to

Here is a continuous variableis additional energy is large except near

^ ' 2 21[ ] ( ) ( )2 c c r cr

c c

H J Uσ σ σ σ+= − +∑∑ ∑cσ

2( )cU σ2( )cU σ 1cσ = ±

From Ising model to the Ginzburg-Landau model

Fig.1 A sharp minimum of at 1 implies that the magnitude of is effectively restricted to nearly 1.

2( ) /cU Tσ

cσ

From Ising model to the Ginzburg-Landau model

The XY model and Heisenberg modelIf the spins are not restricted to point along one axis, we need to describe each spin by a vectorFor Heisenberg model

For XY model

1 2 3( , , )c c c cσ σ σ σ=

1 2( , )c c cσ σ σ=

From Ising model to the Ginzburg-Landau model

Cell HamiltonianDescribe interactions between cell spinsParameters sum up the relevant effects of the details within a scale smaller than a unit cell

Block HamiltonianDescribe interactions between block spins,eachblock consists of unit cells. i.e. Thus parameters in Block Hamiltonian sum up the relevant details within a scale of lattice constants

db 2, 3b or=

b

From Ising model to the Ginzburg-Landau model

Construction of block spinsLabel the blocks by the position vector x of the centers of the blocksDefined as the net spin in a block

The sum in the second equation is in the first Brillouin zone

/2

( )

xdx cc

d ik xk

k

b

x L e

σ σ

σ σ

−

− ⋅

<Λ

=

=

∑

∑

From Ising model to the Ginzburg-Landau model

Construction of block spinsBlock spin is simply the mean of the cell spins within the block labeled by x

That is to say the spatial resolution of the block Hamiltonian is b, whereas that of the cell Hamiltonian is 1

From Ising model to the Ginzburg-Landau model

Ginzburg-Landau formStarts by assuming a simple form for the block Hamiltonian

The coefficients are functions of T, and h is the applied magnetic field divided by T.

2 4 20 2 4

2 2

12

2

1 1

[ ] / [ ( ) ]

Where

( ) ( ) ( ( ))

( )

d

n

ii

d ni

i

H T d x a a a c h

x x x

xα α

σ σ σ σ σ

σ σ σ σ

σσ

=

= =

= + + + ∇ − ⋅

≡ ⋅ ≡

⎛ ⎞∂∇ ≡ ⎜ ⎟∂⎝ ⎠

∫

∑

∑∑

0 2, 4, ,a a a c

From Ising model to the Ginzburg-Landau model

Ginzburg-Landau formFourier transformationRelations between Fourier components and the spin configuration

kσ( )xσ

2/

2/

( )

( )

d d ik xk

d ik xk

k

L d xe x

x L e

σ σ

σ σ

− − ⋅

− ⋅

=

=

∫

∑

From Ising model to the Ginzburg-Landau model

Ginzburg-Landau formFourier transformation

Write in terms of

The sum over y is taken over the 2nd nearest neighbors of the block x

20 2

/24 ' '' ' '' 0

' ''

[ ] / ( )

+ ( )( )

dk k

k

d dk k k k k k

kk k

H T a L a ck

L a L h

σ σ σ

σ σ σ σ σ

−<Λ

−− − −

<Λ

= + ⋅ +

⋅ ⋅ − ⋅

∑

∑xσ

2'2 4 2

0 2 4[ ] / ( )2

dx x x x y xy

x

bH T b a a a c hσ σ σ σ σ σ−

+

⎡ ⎤= + + + − − ⋅⎢ ⎥

⎣ ⎦∑ ∑

From Ising model to the Ginzburg-Landau model

Meaning of various termsObviously is the external field termIf we drop the term and set h=0, we can see that

Each of the terms depends only on of one block, that means each block spin is statistically independent of other block spins.Then we have a system of non-interacting blocks

xh σ− ⋅2( )x x yσ σ +−

2 40 2 4( )x x xU a a aσ σ σ= + +

xσ

/d dL b

From Ising model to the Ginzburg-Landau model

Meaning of various termsWe have no information about the coefficients except they must be smooth functions of and other parametersIt will make no sense that if is negative because would the approach as and the probability distribution would blow up

T

4a( )xU σ −∞ xσ →∞

exp( [ ] / )P H Tσ∝ −

From Ising model to the Ginzburg-Landau model

Meaning of various termsWhen the gradient term is included, then the block spins are no longer independent.This term means interaction between neighboring blocksFor ferromagnets, this interaction will make a block spin parallel to its neighboring block spins.The greater the difference among the block spins, the lager becomes and hence the smaller the probability

2( )x x yσ σ +−

/H T

Landau Theory

Most probable value and Gaussian Approxi.Consider a field and it HamiltonianThe probability of the field isThe most probable configuration is given by

Near can be approximated by

Taylor expansion with

ϕ ( ) /H kTϕ/H kTp e−∝

0ϕ[ ] 0H ϕϕ

∂=

∂

0 , /H kTϕ2

0 02

1[ ] / [ ] / ( )2

H kT H kTϕ ϕ ϕ ϕλ

= + −

0

22 1

2

HTϕ ϕ

λϕ

− −

=

⎛ ⎞∂= ⎜ ⎟∂⎝ ⎠

Landau Theory

Most probable value and Gaussian Approxi.The probability is approximated as

Partition sum

Free energy

Generalize to more degrees of freedom is straightforward

20 0[ ] ( )/2He ϕ ϕ ϕ λ− − −

20 0

0

[ ] ( )/2

[ ] 1 2

H

H

Z d e

e

ϕ ϕ ϕ λ

ϕ

ϕ

πλ

− − −

−

=

= +

∫

20

1ln [ ] ln(2 )2

f Z H ϕ πλ= − = −

Landau Theory

Minimum of Ginzburg-Landau HamiltonianAt the minimum point field is constThat means the Fourier components

The value can be found by setting

We get

0( )xϕ ϕ=

/20 00 for 0 and d

k k Lσ σ ϕ−= ≠ =

0

0

[ ] 0H ϕϕ

∂=

∂

32 0 4 02 4 0a a hϕ ϕ+ − =

Landau Theory

Minimum of Ginzburg-Landau HamiltonianWhen When

When

2 0 20 , / 2a h aϕ> ≈

2 0a <1/2

0 0 2 40 ( / 2 )h m a aϕ= ⇒ = = ± −2

0 0 0 40 | | / (8 )h m h m aϕ> ⇒ = +

2 0a =1/3

044

ha

ϕ⎛ ⎞

= ⎜ ⎟⎝ ⎠

Landau Theory

Minimum of Ginzburg-Landau HamiltonianHere is denoted by 0ϕ σ

Landau Theory

Minimum of Ginzburg-Landau HamiltonianCritical point is atWhen

Energy density

When

2 2 20 '( )ca a a T T= ⇒ = − +

20, 0h a= <2

2 4 20 0 2 0 4 0 0

4

[ ] / ( ) ( )4

d d aH T L a a a L aa

ϕ ϕ ϕ= + + = −

220 0

4

'[ ] / ' ( )2

dc

ae H L a T Ta

ϕ= = − −

2 0 00, 0 'a e aϕ≥ = ⇒ =

0

220

4

' '' ( )

2

c

c c

a T TC aa T T T T

a

>⎧⎪= ⎨ − − <⎪⎩

Landau TheoryMinimum of Ginzburg-Landau Hamiltonian

Calculation of critical exponents

Same as mean-field calculation

( )

2 01/2

22 0

4

1/32 0 4

2 0 22

2 0 0 22

00, 0

' 1/ 2'( )0,2

0, / 4 1/ 3

10, / 2 12 '( )

10, | | / 4 ' 14 '( )

c

c

c

ha

a T Taa

a h a

a h aa T T

a m h aa T T

ϕ

βϕ

ϕ δ

ϕ χ γ

ϕ χ γ

=

> =⎧⎪

⇒ =⎡ ⎤⎨ −< = ⎢ ⎥⎪

⎣ ⎦⎩

= = ⇒ =

⎧ > = ⇒ = ⇒ =⎪ −⎪⎨⎪ < = − ⇒ = ⇒ =⎪ −⎩

Chap. 7 Gaussian approximation for the Ginzburg-Landau

model

Youjin Deng09.12.12

Gaussian approximation for T>Tc

We set h=0 for simplicity, thus

Partition sum

Free enegry

0 0ϕ = 0 0[ ] / dH T a Lϕ =2

0 2[ ] / [ ] / ( ) k kk

H T H T a ckϕ ϕ σ σ−<Λ

≈ + + ⋅∑2 2

20

0

( )| |[ ]/

1/2

22

kk

d

a ckH T

k

a L

k

Z e d e

ea ck

σϕ σ

π

+−

−

∑=

⎛ ⎞= ⎜ ⎟+⎝ ⎠

∫

∏

20 2

1 ln[ / ( )]2

d

kf a L a ckπ−= − +∑

Gaussian approximation for T>Tc

Let , thus

2 22

1( ) | | ( )2kG k a ckσ= = +

cT T= 2 0a =

2

1

0

( ) 0lim ( ) ( ) 1ck

G k kG k T T

η

χ γ

−

−

→

∝ ⇒ =

∝ ∝ − ∝ =

Gaussian approximation for T>TcEnergy density and specific heat

For are singular!!Let

2 12

2

2 22

2

( )

( )

d

d

fe d k a ckaeC d k a cka

−

−

∂∝ ∝ +∂∂

∝ ∝ +∂

∫

∫2 1 2 2

2 2 2, 0, ( ) , ( )a k a ck a ck− −→ + +

1 1/22 2'/ , ( / ) 0 if 0k k a c aξ ξ −= ≡ → →

2 2 4 4 (2 /2)'(1 ' ) ( )

2 / 2

d d d dcC d k k T T

d

ξ ξ

α

− − − − −⎡ ⎤∝ + ∝ ∝ −⎣ ⎦⇒ = −

∫

Gaussian approximation for T<TcHamiltonian of Ginzburg-Landau form

We expand this Hamiltonian for T<Tc near Let use the Taylor expansion, note that

Then we get

2 4 20 2 4[ ] / [ ( ) ]dH T d x a a a h cϕ ϕ ϕ ϕ ϕ= + + − − ∇∫

0ϕ2 4

0 2 4( )f a a a hϕ ϕ ϕ ϕ= + + −

1/2

20 0 2 0

4

/ 4 ,2am h a ma

ϕ⎛ ⎞−

= − = ⎜ ⎟⎝ ⎠

0

2 242 0

2

[ ] / [ ] /

23 [( 2 )( ) ( ) ]2

d

H T H T

ad x a h ca

ϕ ϕ

ϕ ϕ ϕ

≈

+ − + ⋅ − − ∇−∫

Gaussian approximation for T<TcFourier transformation and let

Thus

And free energy

Specific heat

42

2

2322

ab a ha

= − + ⋅−

2 20[ ] / [ ] / ( ) | |d

kH T H T d k b ckϕ ϕ σ≈ + +∫2 2 1

2 12

( ) | | ( )

when 0, ( ) [2 ( ) ]k

c

G k b ck

h G k a T T ck

σ −

−

= ∝ +

= ∝ − +

20

1[ ] / ln[ / ( )]2

d

k

f H T L b ckϕ π−= − +∑

4 dC ξ −∝

Correlation length and temperature dependence

The singular temperature dependence of the quantities can be summarized in terms of correlation length

Where and are both constants fixed by previous calculatiom

2/c aξ =2

2 2

40

2

2 2

40

( )2(1 )

( )4(1 / 2)

'

c

d

c

d

cT T G kk

C C

cT T G kk

C C

ξξ

ξ

ξξ

ξ

−

−

> =+

= +

< =+

= +

0C 0 'C

Correlation length and temperature dependence

Correlation length measures the distance over which spin fluctuations are correlatedSingular behavior of quantities for vanishing

and can be viewed as a result of.

As far as the singular temperature dependence is concerned, is the only relevant length.

| |cT T− kξ →∞

ξ

Ginzburg Criterion

The importance of fluctuationLet’s see the ratio

is the const fixed by calculation

When the temperature is close to Tc within a range , the fluctuation are expected to be important

The smaller is ,the smaller this range will be

4 2 /20 / [ / |1 / |]d d

T cC C T Tξ ζ− −Δ −∼0C

2/(4 )0

1/20 2

[(2 ) / ]

( / ' )

d dT

c

C

c a T

ζ πξ

ξ

− −= Δ

≡

T cTζTζ

Chap. 8 Renormalizaion group

Youjin Deng09.12.12

MotivationStudy of symmetry transformations are proven to be extremely usefulAt the critical region, we want to ask under what S.T is a system invariantMicroscopic details seem to make very little difference in critical phenomena suggests there’s some kind of symmetry properties.This desired transformations is know as renormalization group

MotivationStudy of symmetry transformations are proven to be extremely usefulAt the critical region, we want to ask under what S.T is a system invariantMicroscopic details seem to make very little difference in critical phenomena suggests there’s some kind of symmetry properties.This desired transformations is know as renormalization group

Variable transformation in probability theoryBefore we go to the Kadanoff transformation, first have a reviewProbability distribution , x and y are two random variables, define then the probability distribution for z

does exactly the same job as ,as far as he average values involving z are concerned

( , )P x y1/ 2( )z x y= +

1'( ) ( ( )) ( , )2

1 ( ( ))2 P

P z dxdy z x y P x y

z x y

δ

δ

= − +

= − +

∫

'( )P z ( , )P x y

Variable transformation in probability theoryExample: are identical probability distribution for Now we calculate the distribution for For a integer ,we get

( ) and ( )P x P y, 1, 2,3, 4,5,6,7,8,9,10x y =

z

0 0,

( ) ( , ) ( ) ( )x y

P z z z x y P x P yδ= = +∑0 [2, 20]z ∈

1234567891020191817161514131211987654321

1098765432z( ) 100P z ×

z( ) 100P z ×

Kadanoff transformationTransform cell Hamiltonian to block Hamiltonian

Where are the block spin and cell spin, respectively and the index c runs over all cells and i,j from 1 to n (number of components)b indicates the block size is b times the cell sizeStill we can construct another block Hamiltonian

^

^

[ ]/ [ ]/,

, ,

[ ] / [ ] /

( )

b

xH T H T dix ic j cc

i x j c

H T K H T

e e b dσ σ

σ σ

δ σ σ σ− − −

=

= − ∑∏ ∏∫

''[ ] / [ ] /sH T K H Tσ σ=

x candσ σ

Kadanoff transformationWe can see

In general

Kadanoff transformations will play a major role in the construction of the renormalization group

^ ^[ ] / [ ] / [ ] /s s b sbK H T K K H T K H Tσ σ σ= =

' 's s ssK K K=

Definition of renormalization groupWe take GL form for instance with h=0

Use the triplet of parameters to label the probability distributionsParameter space: different values of parameters and every probability distribution is represented by a point in this parameter space

'2 4 2 22 4

1[ ( ) ]2

H

dx x x x yy

x

P e

H b u u b cσ σ σ σ

−

−+

∝

= + + −∑ ∑

2 4( , , )u u cμ =

Definition of renormalization groupWe define a transformation by the following steps

1,Apply the Kadanoff transformation

This step downgrades the spatial resolution of spin variations to sb, note that is the block Hamiltonian

2,Relabel the block spins in and multiply each of them by a constant

Now the block size sb is shrink to b, back to original

' sRμ μ=

''[ ] [ ]sH K Hσ σ=

[ ]H σ

xσ ''[ ]H σ

sλ'[ ] ( ''[ ])

' /x s x

H H

where x x sσ λ σσ σ →=

=

Definition of renormalization groupThese two steps can be explicitly written as

Write in the GL form

New parameters

Thus define

'[ ]/ [ '']/

'

( '') ''xH T H T ds x y yy

x y

e e s dσ σ δ λ σ σ σ− − −= − ∑∏ ∏∫

'H2 2 2 4

' ' ' 2 ' 4 '''

1'[ ] [ ' ( ) ' ' ]2

dx x y x xy

xH b c b u uσ σ σ σ σ−

+= − + +∑ ∑

2 4' ( ', ', ')u u cμ =

sR

Definition of renormalization groupThe set of transformations is called RGIt’s a semi-group, not a group since the inverse transformations are not defined.It has the property

only if where a is independent of s.

{ , 1}sR s ≥

' 's s ssR R R=

as sλ =