Spontaneous magnetization for H=0
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Transcript Spontaneous magnetization for H=0
Ginzburg-Landau theory of second-order
phase transitions
Second order= no latent heat
(ferromagnetism, superfluidity, superconductivity).
Let the transition occur at T=TC
Ordered phase below transition (typically)
Order parameter h (vanishes in disordered phase):
h = macrospopic wave function (superconductivity)
h = magnetization (ferro or antiferro)
Vitaly L. Ginzburg
Example: liquid crystals
Lev Landau
Order parameter
h = excess population at angle Q (or a simple function of Q )
in nematic-isotropic transition in liquid crystals-simplest example (1 real scalar parameter)
One of the most common LC phases is the
nematic. The word nematic comes from
the Greek νημα (nema), which means
"thread”
Alignment in a nematic phase
.
1
(see lectures by A. Salonen)
Tfy-0.3252 Soft Matter Physics, Fall 2009 / E. Salonen
2
Equilibrium Condition
G (h ) U PV TS
Gibbs free energy
G
0
h
T
Gibbs free energy must be
minimum at equilibrium at any T
dG SdT VdP
2G
2 0
h T
How G depends on h near the transition Temperature
Order parameter h is 0 or small (since it vanishes in disordered phase)
Near the transition
G G0 nh n , G0 disordered phase free energy
n
Gibbs free energy must be minimum at
equilibrium in disorderesìd phase
G
0
1 2 2h ... 1 for h 0.
h
T
Therefore the expansion around critical point starts from second order
G G0 2h 2 3h 3 4h 4
At h=0 there is a minimum or a maximum
3
GL Phenomenological model, with terms up to 4° order:
G G0 2h 2 3h 3 4h 4
disordered 2 (T ) 0, T TC
G-G0
At the transition T= TC
G must be the same for
nematic and isotropic
(equilibrium).
h
ordered 2 (T ) 0
Above TC minimum at h=0 (order parameter vanishes at equilibrium in disordered phase
2G
2 0 2 2 63h 0 2 (T ) 0
h T
2 (T ) 0 taken linear
Below TC maximum at h=0
1
2
2 (T ) (T T * ) A0
parameter T * usually different from TC : T * TC
G G0 (T T * )
A0 2
h 3h 3 4h 4 , 4 0 in order to have overall minimum
2
4
1
G G0 (T T * ) A0h 2 3h 3 4h 4 ,
2
1
1
1
often written in the form
G G0 A0 (T T * )h 2 Bh 3 Ch 4
2
3
4
Discontinuity of Order Parameter h at Tc
One obtains the discontinuity of order parameter by imposing equilibrium and minimum at Tc
Equilibrium : G G0
1
1
1
1
1
1
A0 (TC T * )h 2 Bh 3 Ch 4 0 A0 (TC T * ) Bh Ch 2 0
2
3
4
2
3
4
G
1
1
1
*
2
3
Minimum : G G0 (T T * ) A0h 2 Bh 3 Ch 4 0
A0 (T T )h Bh Ch
2
3
4
h T
G
0
h T
G G
0
A0 (TC T * ) Bh Ch 2 0
1
1
1 2
*
A
(
T
T
)
B
h
Ch 0
0 C
2
3
4
Divide the first by 2 and subtract:
hC
2B
3C
B
C
h h 2 0 and find hc
6
4
Choosing h 0, B 0 since C 0 (minimum condition)
5
hC
T*
2B
3C
Choosing h 0, B 0
One can write T* in terms of TC and the other parameters. The order
parameter is discontinuous at transition.
1
1
1
G G0 A0 (T T * )h 2 Bh 3 Ch 4
Insert hhc back into G-G0 equation
2
3
4
1
1
1
G G0 at thetransition h 2 ( A0 (TC T * ) Bh Ch 2 ) 0
2
3
4
1
1
2B 1
2B 2
A0 (TC T * ) B (
) C (
) 0
2
3
3C 4
3C
2 B2
2 B2
*
*
get A0 (TC T )
TC T
9 C
9 CA0
Order parameter versus T
Above, we got the discontinuity
hC
2B
3C
To obtain h(T) we can start from the equilibrium condition
G
*
2
3
0
A0 (T T )h Bh Ch
h T
nematic : h (T )
B B 2 4 A0C (T T * )
2C
6
To get rid of the double sign, choose the right h at the transition:
insert
B B 2 4 A0C (T T * )
2 B2
TC T
into h (T )
9 CA0
2C
*
2 B2
B2
B B 4C
B
9 C
9 , recalling minimum condition B 0.
h
2C
2C
|B|
B 2 B OK with discontinuity h
B
B
c
3C
3
3
h
B
2C
2C
wrong
3C
2
we choose h (T )
B B 2 4 A0C (T T * )
2C
: order parameter versusT .
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Langevin paramagnetism
Model: "gas" of ions with magnetic moments g B J
H g B J .B magnetization M
N z
Volume
Quantum partition function
Z
J
m J
mg B B
exp[
]
K BT
Paul Langevin
(1872-1946)
B
1
In the special case J , with g 2 find Z 2 cosh( B )
2
K BT
B H
N
M B tanh
V
K
T
B
8
Weiss mean field theory (1907) of Ferromagnetism:
Idea:the effective field gets a contribution
from the magnetization: H eff H M ,
B H
1
N
λ to be determined J , g 2 M B tanh
2
V
K BT
B ( H M )
N
M B tanh
V
PIERRE-ERNEST WEISS
born March 25, 1865, Mulhouse, France.
died Oct. 24, 1940, Lyon, France.
K BT
The spontaneous magnetization may remain even for H=0
M
M
N
B tanh B
V
K
T
B
(Ferromagnetism!)
One can fix the parameter in terms of Tc as follows:
M B M
with H 0 since M small, tanh B
K BT
K BT
2
K BTCV
B M
MV
N
B
.
TC
is very useful.
2
N B
K BTC
N B
VK B
For T TC
9
M
N
B tanh B
can be solved graphically.
V
K BT
M
N
x B
M B tanh x . Graphical solution:
K BT
V
M
Set
tanh x
B M B
T
K
K
T
M
M B B
N
N
TC
B
B B
V
V
KB
N B2
xT
, TC
TC
VK B
x=y
1 solution if T>Tc,
2 otherwise
M
M
N
B tanh B
can be solved graphically.
V
K BT
From the graphical solution we gain the trend with T of the M(T) curve.
Increasing T at H=0 M must vanish, but how?
PIERRE-ERNEST WEISS
Spontaneous magnetization for H=0
born March 25, 1865, Mulhouse, France.
died Oct. 24, 1940, Lyon, France.
T Tc from below:
M is very small x M
xT
x 3 xT
tanh x
. We need the third power: x
TC
3 TC
M x
K BT
B
M
B
K BT
x 3(1
1
T
)
TC
TC T . Power law!Critical exponent 1/2.
11
A different experiment: fix T =Tc and measure M dependence on H at
for H0 . Since M is small we can expand
B (H M )
N
M B tanh
V
K
T
B C
3
B (H M ) 1 B (H M )
N
B [
...]
V
K BTC
K BT
3
N B2
N
M
(H M ) B B
VK BTC
V
K BT
N B2
Using TC
, the
VK B
We are left with:
3
3
2
2
2
3
3
( H 3H M 3H M M )
N B2
l.h.s simplifies with
M .
VK BTC
3
B
N
N
3
2
2
2
3
3
0
H B
(
H
3
H
M
3
H
M
M
).
VK BTC
V
K
T
B
2
B
1
Setting M H
for
H 0, neglecting H 2 and H 3
3
1
B 3 3
N
N
3
3
0
H B
M M H 3. M H .
VK BTC
V
K BT
Then, neglect of H 2 and H 3 justified a posteriori.
2
B
Power law!Critical exponent 1/3.
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Behavior for T >Tc , in paramagnetic phase M small
B ( H M )
N
M B tanh
V
K
T
B
N B 2
(H M )
V K BT
This gives
N
M
V
B 2
B 2
N B2
N
H
H , TC
2
N B
V K B (T TC )
VK B
K BT
V
which is the Curie law. Critical exponent=-1.
Of course this holds in para phase while M is small.
How do these critical exponents arise? Widom in 1965 put forth the scaling
hypothesis for magnetic materials in field B:
G Gibb's free energy: r , p : G ( r (
T
T
), p B) G(( ), B) G homogeneous.
TC
TC
f homogeneous function a,b:f( a x, b y) f ( x, y)
This produces relations between various critical exponents.
The dependence on approximation scheme and dimensionality is dramatic.
(Weiss theory)
(Weiss theory)
14
Ising Model in 1d , defined on a closed ring
(model invented in 1920 by Wilhelm
Lenz as a thesis for Ernst Ising)
H Ising Jsn sn 1 H sn
n
H H
Simplify notation :
n
H Ising Jsn sn 1 H sn
n
n
where the sums over sites and sn is a classical spin taking the values +1,-1.
The Partition function is defined by:
Z Tre
H Ising
From Z one derives thermodynamics: F KT log Z .
Z e
s1 s2
J ( s1s2 s2 s3 sN s1 ) H ( s1 s2 sN )
e
.
sN
15
Z e J ( s1s2 s2s3 sN s1 ) e H ( s1 s2 sN ) .
s1 s2
Rearrange:
sN
Z [e Js1s2 e Hs2 ][e Js2 s3 e Hs3 ] [e JsM s1 e Hs1 ].
s1 s2
sN
Vs1s2 [e Js1s2 e Hs2 ]
1
2
There is a [ ] factor for each bond; each factor is a matrix product
Jsn sn1 Hsn1
(VV
)
(
V
)
(
V
)
[
e
]
1 2 sn , sn1
1 sn , s
2 s , sn1
where V VV
1 2
called transfer matrix
(V1 ) s , s e Jss
(V2 ) s , s e Hs ( s, s)
e J
V J
e
e J e H
e J 0
0 e ( J H )
( J H )
H
e
e
Z [VV
1 2 ]s1s2 [VV
1 2 ]s2 s3
s1 s2
sN
e ( J H )
e ( J H )
[VV
1 2 ]sM s1 .
16
Z [VV
1 2 ]s1s2 [VV
1 2 ]s2 s3
s1 s2
sN
[VV
1 2 ]sM s1 . But pbcimply a Tr :
M
M
Z [(VV
)
]
TrV
, V VV
1 2
s1s1
1 2
called transfer
matrix
s1
V V
V
V
V
recall : Z [e Js1s2 e Hs2 ] [e Js2 s3 e Hs3 ] [e JsM s1 e Hs1 ].
s1 s2
e J
V J
e
sN
e J e H
J
e 0
0 e ( J H )
( J H )
H
e e
e ( J H )
( J H )
e
e ( J H )
e ( J H )
Det[V I ] ( J H )
0
( J H )
e
e
2 e J (e H e H ) e 2 J e 2 J 0
2 2 e J cosh( H ) 2sinh(2 J ) 0
17
2 2 e J cosh( H ) 2sinh(2 J ) 0
e J cosh( H ) e2 J cosh( H )2 exp(2 J ) exp(2 J )
e J cosh( H ) e J cosh( H )2 1 exp(4 J )
e J cosh( H ) e J sinh( H )2 exp(4 J )
For H0,
J
e e
J
exp(4 J ) e
TrV
TrV N N N
J
J 2 J
e e
e
J
e
J
2cosh( J )
2sinh( J )
Z N N .
18
We need to note that another reasoning is possible. Instead of
Vs1s2 [e Js1s2 e Hs2 ]
1
2
one can decide to distribute the interactions with H symmetrically among the bonds and
define a different transfer matrix, which is also common in textbooks, with elements:
1
Vsl , sm e
1
2
Vs2 , s2 [e Js1s2 e
2
Jsl sm H ( sl sm )
,
e ( J H )
V J
e
1
H ( s1 s2 )
2
]
e J
e ( J H )
This is transfer matrix has the same trace and the same determinant as the previous one,
hence the eigenvalues are the same.
19
Thermodynamic limit: only large eigenvalue matters
Z N N N
e J cosh( H ) e J sinh( H ) 2 exp(4 J )
F NKT ln NJ NK BT ln cosh( H ) sinh( H ) 2 e 4 J .
F
sinh( H )
Magnetization (after simplification)
.
2
4
J
H
sinh( H ) e
No spontaneous magnetization.
The peak in specific heat is
due to some short-range
order (increasing T at low
temperatures spins are
reversed and this costs
energy).
20
no ferromagnetism, no phase transition, no long-range order!
1d theory fails to explain magnetism. Why?
Ground
state :
1 broken bond :
Breaking a single bond breaks long range order. It costs energy 2J.
S=K B logW.
The free energy change is
F 2 J K BT ln( N )
For large N there is a gain in F at any T. In 2d is different.
21
Rudolf Peierls Centre for Theoretical
Physics
1 Keble Road
Oxford, OX1 3NP
England
22
Percolation problem: Renormalization group approach to the
onset of conductivity
A.P.Young and R.B. Stinchcombe J. Phys. C 8 L535 (1975)
Consider a regular lattice (sites connected by bonds, with Z bonds per site)
Remove a fraction 1-p of bonds at random. Conductivity is a function of p.
There is a critical value pc such that for p< pc the lattice is not conducting
and consists of clusters of connected sites in a sea of isolated sites. If we
pick two sites far apart the probability of finding a continuous path joining
them gets small. Conversely, for p> pc the film as a whole is conducting ,
even if it contains nonconducting islands. The macroscopic conductivity S
of the lattice is a function SS(p, s, s bond conductivity .
Let us consider a square lattice:
How to find an approximation to pc ?
By scaling!
For exampe, discard half of the x and
half of the y and consider a new lattice
where the nodes can be connected or
not with some probability. The new
problem is similar. We can exploit that!
23
Decimation
Rescaled length (rescaling factor b )
Rescaled
probability
p pc p1 ( p, b) 0
p1 p1 ( p, b)
for b
p pc nonconducting islands disappear
p pc system unchanged
for b
for b p1 p1 ( pc , b )
24
A similar problem can be posed on the lattice with vertices 1,2, .. or
with the rescaled lattice A,B,…on a larger square mesh.
p1= probability in rescaled network , s1 bond conductivity in
rescaled network
SS(p1, s1
25
in this square lattice case:
rescaling factor b 2,
Renormalization of
bond probability :
p1
p
at each scaling
p changes by a
factor ( p).
Rescaled length (rescaling factor b 2 ),
the correlation length index
b 2
1
is defined by b c
ln(b) ln( 2)
0.8547
ln( ) ln( 3 )
2
By comparison with the exact results one can evaluate approximations.
To proceed find a simple approximation to p1 p1 ( p, b)
Consider only paths involving ,,c,.
What is the probability p1 that nodes A and B of new lattice are connected?
1
B
A
Let
2
D
P( , , , ) probability that , , ,
C
are connected,
P( , , ) probability that α,β,γ are conected but δ is not, and so on.
p1 P( , , , ) P( , , ) P( , , ) P( , , ) P( , , ) P( , ) P( , )
renormalization of p :
p1 p 4 4 p 3 (1 p) 2 p 2 (1 p) 2 2 p 2 p 4
27
Renormalization of
p1 2 p 2 p 4
bond probability :
p1 4 p (1 p 2 ) p
p1
4 p (1 p 2 )
p
at each scaling p changes by a factor ( p).
p1 2 p 2 p 4 Fixed point equation:
pc 2 pc 2 pc 4
Fixed points
x 2x2 x4
x 0 or
x 3 2 x 1 0 which yields
Roots : x 0, x 1, x
x 1 or
x2 x 1 0
5 1
5 1
0.618, x
1.618
2
2
pc 0 Trivial Fixed point : repeated scaling no conductance
pc 1 Trivial Fixed point : repeated scaling perfect conductance
pc 1.6 nonsense fixed point
nontrivial solution :
pc 0.618 (Exact
result
pc 0.5)
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