"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

'qq

,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