"local" landau-like term gradient term taking care of fluctuations free energy of...
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"local" Landau-like term
gradient term taking care of fluctuationsfree energy of
disordered phase
("effective Hamiltonian")cost involved in creating
inhomogeneities
V
tCtc
V
tBtb
)()( ,
)()(
average value of order parameter = 0 in disordered phase = 0 in ordered phase
contribution from fluctuations
,2
2/1
0
1
c
tbcTT
01 a
Ginzburg criterion
Levanyuk criterion
d: dimensionality
T=Tc: fractal structure
fluctuations of all length scales possible
no typical length scale
’’ (H’, T’)
majority rule
homogeneity property
close to the critical point!
dimensionality
EJERCICIO 15
In practice, this only works near the critical point. At the critical point does not change on RG transformation!
The renormalisation group exploits properties at and near T=Tc
Renormalisation group transformation
H=0
RG
the K0 parameter is
needed!but only K1 and K2 are relevant
=0=
repulsive
attractive
mixed
=0T=0 orT=
NON-TRIVIAL FIXED POINT
=0=
to trivial fixed pointT=0
to trivial fixed pointT=
(points with= )
(point with= 0)
(point with= 0)
k
=K’=K*+k’
SCALING FIELDS
UU '
UUU 2'" k Α'k U
20
0'
1U
…
U1U2
diagonalise
renormalisation group
Therefore we can write:
yl
ywhere are some exponents
some are positive (flow away from the critical surface)
increase with iterationsthe others are negative (flow on the critical surface) decrease with iterations
UlU y'
In the coordinate frame where A is diagonal the RG transformation is very simple:
With all of this, it is easy to accept the scaling behaviour
This implies that all critical exponents can be obtained from y1,y2
),(),( 212121 UlUllUU yyd
k
yyy lll 22
RG
1.721 -0.387
3
210,11
3
210,12
1
2
12
21212
2
11
2211
2211
3
2
9
1
3
1'
9
1
3
4
3
1
3
22
9
1
9
2
9
1
3
12'
3
1
'9
1' ,'
3
1'
9
1 ,
3
1
kkk
kkkkkkk
kKkK
kKkK
Linearisation:
1
2
Ejercicio 16:
k
2111 3
210kkkU
2122 3
210kkkU
The critical line of the problem is given (linear approx.) by:
3
102
9
10312
KK
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