Landau Theory of Phase Transition 11-21-18bingweb.binghamton.edu/~suzuki/ThermoStatFIles/16.3 PT...
Transcript of Landau Theory of Phase Transition 11-21-18bingweb.binghamton.edu/~suzuki/ThermoStatFIles/16.3 PT...
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