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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 =