Landau Theory of Phase Transition

Masatsugu Sei Suzuki

Department of Physics, SUNY at Binghamton

(Date: November 29, 2017)

Lev Davidovich Landau (January 22, 1908 - 1 April 1968) was a Soviet physicist who made

fundamental contributions to many areas of theoretical physics. His accomplishments include the

independent co-discovery of the density matrix method in quantum mechanics (alongside John

von Neumann), the quantum mechanical theory of diamagnetism, the theory of superfluidity, the

theory of second-order phase transitions, the Ginzburg–Landau theory of superconductivity, the

theory of Fermi liquid, the explanation of Landau damping in plasma physics, the Landau pole in

quantum electrodynamics, the two-component theory of neutrinos, and Landau's equations for S

matrix singularities. He received the 1962 Nobel Prize in Physics for his development of a

mathematical theory of superfluidity that accounts for the properties of liquid helium II at a

temperature below 2.17K.

https://en.wikipedia.org/wiki/Lev_Landau

The theory of phase transitions of second order is based on the thermodynamic properties of

the system for given deviations from the symmetrical state. Landau introduced the concept of the

order parameter. The thermodynamic properties can be expressed in terms of the order parameter

.

Fig. (a) Mean-field susceptibility, (b) specific heat, (c) correlation length, and (d) order

parameter as a function of temperature. The mean field exponent; 0' . 2

1 .

1' . 3 . 0 . 2

1 .

We consider the Helmholtz free energy; ),( VTFF .

STUF

is a minimum in equilibrium. This free energy is a minimum with respect to an order parameter

.

((Example))

Magnetization, dielectric polarization, fraction of superconducting electrons in a

superconductor, etc.

In thermal equilibrium, will have a certain value )(0 T . In the Landau theory, we imagine

that can be independently specified. We consider the Landau-free energy function

),(),(),( TTSTUTFL

where

),(),()( 0 TFTFTF LLL

if 0 .

),( TFL is an even function of , in the absence of applied fields.

Fig. Plot of the free energy as a function of the order parameter

((Assumption))

)(LF is an even function of in the absence of applied fields.

...)(4

1)(

2

1)(),( 4

4

2

20 TgTgTgTFL

For simplicity,

)()(2 cTTaTg

const)(4 Tg >0, 0)(6 Tg .

Then we have

4

4

2

04

1)(

2

1)(),( gTTaTgTF cL

and

0)( 3

4

gTTF

cL

which has the roots

0

and

)(4

2 TTg

ac

or

2/1

2/1

4

)( TTg

ac

(

2

1 )

with

0a and 04 g

where is the critical exponent of magnetization.

(i) The root 0 corresponds to the minimum of FL at T above Tc.

)()( 0 TgTFL .

(ii) The other root )(4

2 TTg

ac corresponds to the minimum of LF at T above Tc.

2

4

2

0 )()()( cL TTg

aTgTF .

2. Susceptibility

We consider the case when an external magnetic field is present.

BgTTaTgTF cL 4

4

2

04

1)(

2

1)(),(

in the presence of external field B.

0)(),( 3

4

BgTTa

TFc

L

BgTTa c 3

4)(

Taking derivative of

),( TFL with respect to B;

1]3)([ 2

4

B

gTTa c

The susceptibility is given by

2

43)(

1

gTTaB c

For cTT , 0

)(

1

cTTaH

)1(

where is the critical exponent of the susceptibility.

For cTT

2/1

2/1

4

)( TTg

ac

. (

2

1 )

)(2

1

)(3)(

1

TTaTTaTTa ccc

( )1'

Fig. The order parameter ( )0(/)( 0 t as a function of reduced temperature, cTTt / .

2/1

4

)0(

g

aTC .

Fig. The reduced susceptibility caT as a function of reduced temperature, cTTt / .

t T Tc

t 0

0.2 0.4 0.6 0.8 1.0 1.2

0.2

0.2

0.4

0.6

0.8

1.0

1.2

t T Tc

aTc t

0.8 1.0 1.2 1.4

5

10

15

20

25

3. Entropy and heat capacity

The entropy S is given by

0

2

0

4

( )

( ) ( )c

S TF

ST

aS T T T

g

)(

)(

c

c

TT

TT

where

20

2

0

( ) 1( )

2

1( ) ( )

2

c

c

FS

T

g Ta T T a

T T

S T a T T aT

with

T

TgTS

)(

)( 00

So that there is no latent heat at the transition temperature T0. Such a transition is called a second

order phase transition.

We can evaluate the heat capacity

V

VT

STC

of the two phases for cTT and cTT .

0

2

0

4

( )

( )

V

dS TT

dT

C

dS T aT T

dT g

)(

)(

c

c

TT

TT

At cTT , the heat capacity is discontinuous (second order phase transition).

2

4

V c

aC T

g (Discontinuity of VC at cT T .

Fig. Schematic plot of the heat capacity as a function of temperature.

4. 1/

M B

At cT T , we have

3

4g B or

1/3

4

B

g

So we have the critical exponent 3.

5. Correlation length

Suppose that the order parameter is not homogenous in space. We add a term proportional to

2 . The free energy is modified as

24

4

2

0

2

)(4

1)(

2

1)(

)(),(),,(

BgTTaTg

TFTF

c

Lr

where H is dependent on the space. We assume the functional Hamiltonian )(K which is defined

by

])(2

1

4

1)(

2

1[)( 24

4

23 BgTTadK cr

To evaluate a derivative of the functional, we introduce

)()( KKK

where )(r is a small perturbation. We expand )( K as

})]([2

1)()(

4

1))((

2

1{)( 24

4

23 BgTTadK cr

and keep only up to linear terms. Then we obtain

)]()([)()(3

4

3 BgTTadKKK cr

Taking the integral (the last term) by part, requiring that 0 at the surface and minimizing the

total free energy

0)(])([23

4

3 rr BgTTadK c

leading to the equation for

0)( 3

4

2 BgTTa c .

_____________________________________________________________________________

((Note))

)()][3

zzyyxxdxdydzd

r

2

2

2

2

|x

dxx

dxxxx

dx

2

2

2

2

|y

dxy

dyyyy

dy

2

2

2

2

|z

dzz

dzzzz

dz

Thus we get

)()(2

2

2

2

2

2

zyxdxdydz

zzyyxxdxdydz

__________________________________________________________________________

((Green function method))

We solve the differential equation

0)( 3

4

2 BgTTa c

using the Green function method. Suppose that 0 , where is the deviation of from each

equilibrium value 0 .

0)(3

040 gTTa c

and

0))(()(3

0400

2 BgTTa c

or

0)()(3

040

2 BgTTaTTa cc

where 0 depends on T,

00 for cTT

0)(2

04 gTTa c cTT .

(a) For cTT

0)(2 BTTa c

or

BTTa c

]

)([ 2

(b) For cTT

0)(22 BTTa c

or

BTTa c

]

)(2[ 2

Here we define the Green function such that

)'()'()( 22rrrr G (Modified Helmholtz)

with

'4

'exp[)'(

rr

rrrr

G ,

where

1

is the inverse correlation length and is the correlation length

)(2

)(

TTa

TTa

c

c

c

c

TT

TT

(The critical exponent for the correlation, 2

1' ).

Fig. The reduced correlation length )(/ taTc as a function of reduced temperature,

cTTt / .

The solution of

)()()( 22 rr

B

is given by

t T Tc

aTct

0.8 1.0 1.2 1.4

1

2

3

4

5

6

)'('4

'exp['

)'(

'4

'exp['

)'()'(')(

3

3

3

rrr

rrr

r

rr

rrr

rrrrr

Bd

Bd

BGd

When )'(rB is a highly localized magnetic field at the origin,

)0'()'( 0 rr BB

we have

00

3

4

)exp()'(

'4

'exp[')( B

r

rBd

r

rr

rrrr

In general, )(r has the form of

2

1)(

dr

r

Note that 1 and 2

1 . Using the scaling relation )2( , we have

0 (critical exponent)

where is the critical exponent for the pair correlation function. Then we have 12 d and

rr

1)(

for d = 3, which is in agreement from the prediction from the mean field theory.

6. Fluctuation

We define the partition function as

)](exp[ KZ

where

])(4

1)(

2

1[)( 24

4

23 HgTTadK cr

Thus we get

ZMKMH

Z

)](exp[)(

or

1 1 1 lnZ ZM

Z B B

where

r3dM

)](exp[ 1

KMZ

M

ZMKMZH

Z 222

2

2

)](exp[)("

The fluctuation is given by

22 2

2

2 2

2

2

2 2

" '1[ ]

1 '( ) '

1ln

M M M

Z Z Z

Z

Z

Z

ZB

The susceptibility is defined by

2

2

1 lnM Z

B B

Then we have

TkM B12

22 2

2

2 2

2

2

2 2

" '1[ ]

1 '( ) '

1ln

M M M

Z Z Z

Z

Z

Z

ZB

7. Spin correlation

We now consider the system of spin 1/2 magnetic atoms. Each atom has magnetic moment

iB ( 1i ). The magnetization is given by

i

iBM

The fluctuation of M is obtained as

i j

jiB

i

iBM

((

)(

2

222

where ji (( is a spin correlation function. Since

TkM B12

, the

susceptibility is related to the spin correlation function as

i j

ji

B

B

Tk

((

2

8. Universality and critical exponents

Critical exponents describe the behavior of physical quantities near continuous phase

transitions. They are universal, i.e. they do not depend on the details of the physical system, but

only on (i) • the dimension of the system, (ii) the degree of freedom such as Ising (n = 1), XY (n

= 2) and Heisenberg (n = 3) (for spin system), and (iii) the range of the interaction.

Fig. Universality class of the critical behavior, which depends on the dimension (d) and the

number of freedom (n). The Onsager point(d = 2, n = 1). The Kosterlitz-Thouless (KT

point) (d = 2, n = 2). The liquid 4He point (d = 3, n = 2).

M.E. Fisher, Rev. Mod. Phys. 46, 597 (1974).

Magnetizion:

)( TTM c ( cTT )

Susceptibility:

)( cTT ( cTT )

')( TTc ( cTT )

Heat capacity:

)( cTTC ( cTT )

')( TTC c ( cTT )

Correlation length:

)( cTT ( cTT )

')( TTc ( cTT )

Critical isotherm:

/1

HM ( cTT )

((Note)) Mean field critical exponent

0' , 2

1 , 1' , 3

0 , 2

1

9. Scaling relation

There are the following scaling relations for critical exponents.

22 (Rushbrooke identity)

d )2( (Josephson identity)

)2( (Fisher identity)

)2(2

1 d

2

2

d

d

'

'

2)1( (Griffiths identity)

)1( (Widom identity)

where d is the dimension of the system.

10. Universality class

http://www.sklogwiki.org/SklogWiki/index.php/Universality_classes

Fig. Diagram of the (d, n) plane showing contours of constant . There is a region where

2/1 , and a nonphysical region of negative .

M.E. Fisher, Rev. Mod. Phys. 46, 597 (1974).

Fig. Diagram showing contours of constant exponent a in the (d, n) plane. The dash-dot

contours indicate negative a; the solid contours are for 0 .

M.E. Fisher, Rev. Mod. Phys. 46, 597 (1974).

11. Example of the first order phase transition Huang 19-3

The nematic liquid crystal used in display can be described by an order

parameter S corresponding to the degree of alignment of molecular directions. In

the ordinary fluid phase )0( S . The transition between the ordinary fluid phase

and a nematic phase can be modeled via the mean-field Landau free energy

432)( cSbSatSSE ,

where 0TTt , and a, b, c are positive constants. The third-coefficient b is usually

small.

(a) Sketch )(SE for range of T, from 0TT , through 0TT to

0TT Comment on the value of S at the minimum of )(SE at

each value of T considered.

(b) What are the conditions for )(SE to be minimum?

(c) Find the transition temperarure Tc. (It is not T0)

(d) Is there a latent heat associated with the phase transition? If so,

what is it?

(e) How does the order parameter vary below Tc?

((Solution))

432)( cSbSatSSE

0)234(432 232

atbScSScSbSatSS

E

where a>0, b>0, c>0.

We make a plot of the free energy E(S) with a = 2, b = 0.8, and c = 0.2

___________________________________________________________________________

For t<0,

E(S) has two local minimum at S = (<0) and S = (>0).

3

2

256

)329()()(

c

actbbEE

with

c

actbb

8

3293 2

and

c

actbb

8

3293 2 .

Note that 0)()( EE

Fig. t = -1.9 (red).

_____________________________________________________________________________

10 8 6 4 2 2 4

100

80

60

40

20

When t = 0, there is one local minimum at .4

3

c

bS , where 3

4

256

27)(

c

bE

___________________________________________________________________________

When 0

2

4TT

ac

bt or

ac

bTTc

4

2

0

3 b

4 c

27 b4

256 c 3

5 4 3 2 1 1

6

4

2

2

Fig. t = 0.38 (red), 0.39, 0.40 (green), 0.41, and 0.42 (purple). The green line at ac

bt

4

2

(in

the present case t = 0.4), where c

bS

2 (=-2).

3 2 1 1

0.2

0.1

0.1

0.2

0.3

0.4

0.5

b

2 c

b

4 c

b4

256 c 3

First order phase transition

3.0 2.5 2.0 1.5 1.0 0.5 0.5

0.2

0.2

0.4

Entropy: 2)(~aS

T

SES

Note that S is no an entropy. The entropy change:

2

2

~

c

baS

The latent heat:

22

02

)4

(~

c

ba

ac

bTSTL

_____________________________________________________________________________

For ac

bt

ac

b

8

9

4

22

Fig. t = 0.44 (red), 0.46, 0.48 (green), 0.50, and 0.52 (purple), 0.54

____________________________________________________________________________

3 2 1 1

0.2

0.2

0.4

0.6

0.8

1.0

When ac

bt

8

92

Fig. t = 0.6 (red), 0.7, 0.8 (green), 0.9, and 1 (purple).

What happens to the order parameter well below Tc?

Fig. t = -1 (red), -0.9, -0.8, -0.7, -0.6, -0.5, -0.4, -0.3, -0.2, -0.1.

3 2 1 1

0.2

0.2

0.4

0.6

0.8

1.0

5 4 3 2 1 1 2

35

30

25

20

15

10

5

_____________________________________________________________________________

REFERENCES

L.D. Landau and E.M. Lifshitz, Statistical Physics (Pergamon Press 1976).

D. Ter Haar edited, Collected Papers of L.D. Landau, p.193 (Gordon & Breach, 1965).

M.E. Fisher, Scaling, Universality and Renormalization group Theory, in F.J.W. Hahne edited

Critical Phenomena (Springer, 1983).

S.K. Ma, Modern Theory of Critical Phenomena (Westview Press, 2000).

H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena (Oxford, 1971).

U. Köbler and A. Hoser, Renormalization Group Theory, Impact on Experimental Magnetism

(Springer, 2010).

P.M. Chaikin and T.C. Lubemsky, Principles of Condensed Matter Physics (Cambridge, 1995).

R.J. Birgeneau, H.J. Guggenheim, and G. Shirane, Phys. Rev.B 1, 2211 (1970).

P. Pfeuty and G. Toulouse, Introduction to the Renormalization Group and Critical Phenomena

(John Wiley & Sons, 1977).

______________________________________________________________________________

APPENDIX-I The first and second phase transitions

t5 4 3 2 1

6

4

2

2

4

____________________________________________________________________________

APPENDIX II Magnetic phase transitions. Typical experimental results

[U. Köbler and A. Hoser, Renormalization Group Theory, Impact on Experimental Magnetism

(Springer, 2010).]

(1) Specific heat of 4He adsorbed on grafoil at critical coverage for 33 epitaxial structure.

(2) Specific heat of GdZn

(3) Sublattice magnetization of K2NiF4

Fig. Neutron scattering intensity at the (1, 0, 0) peak of K2NiF4 as a function of temperature

(Birgeneau et al. 1970).

(4) Sublattice magnetization of CrF2

(5) Sublattice magnetization and staggered suceptibility of USb

(6) Sublattice magnetization of NiF2

(7) UO2

(8) NiCo3

(9) NiO powder

(10) LuFeO3

(11) CrF2

APPENDIX-III Phase transition

L.J. de Jongh and A.R. Miedema, Advances in Physics, 50, 947 (2001). Experiments on simple

magnetic model systems

APPENDIX-II Railroad track analogy of renormalization group

______________________________________________________________________________

Mean field theory

For spin 1/2 system,

tanh( )

tanh( )

B B

B B

NM B

V

n B

or

BM ng S ( 2g , 1

2S ).

We assume that

effB B M

When BN

My

, we have

2

tanh( )

tanh( )

B

B

B

My

n

M

n y

We note that

2

2 BB

B

n yn y y

k T x

or

2

B

B

k Tx

n

The critical temperature is defined by

21 B c

B

k Tx

n

or

2

B c Bk T n

The reduced temperature is

cT

Tx .

So we make a ContourPlot of

)tanh(x

yy

for 0x .

We note that the critical temperature cT is expressed by

2( 1)

3c

B

T zJS Sk

where J is the exchange interaction and z is the number of the nearest neighbor spins. When S =

1/2,

2c

B

zJT

k

Using the relation

2

2B c B

zJk T n

or

22B

zJ

n

((Note))

Spin Hamiltonian:

2 i B eff iH zJ S S g B S

where 2.g The effective magnetic field is

2eff

B

zJB S

g

When g =2 and 1

2S , we have

2eff

B

zJB

which is equal to

22 2B

B B

zJ zJM ng S

n

.

Clear "Global` " ;

f1 ContourPlot y Tanhy

x, x, 0, 1.2 ,

y, 0, 1.2 , ContourStyle Red, Thick ,

PlotPoints 150 ;

f2

Graphics

Text Style "t T Tc", Italic, 12, Black ,

1.1, 0.02 ,

Text Style "y M N B ", Italic, 12, Black ,

0.1, 1.1 ;

f3 Plot 1 t , t, 0, 1 , PlotStyle Blue, Thick ;

Show f1, f2

t T Tc

y M N B

0.0 0.2 0.4 0.6 0.8 1.0 1.2

0.0

0.2

0.4

0.6

0.8

1.0

1.2