"local" Landau-like term gradient term taking care of fluctuations free energy of disordered phase ("effective Hamiltonian") cost involved in creating inhomogeneities
average value of order parameter = 0 in disordered phase = 0 in ordered phase contribution from fluctuations
Ginzburg criterion Levanyuk criterion
d: dimensionality
T=T c : 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=T c Renormalisation group transformation H=0
RG
the K 0 parameter is needed! but only K 1 and K 2 are relevant
=0 =
repulsive attractive mixed =0 T=0 or T= NON-TRIVIAL FIXED POINT =0 =
to trivial fixed point T=0 to trivial fixed point T= (points with = ) (point with = 0) (point with = 0)
k =K’=K*+k’
SCALING FIELDS … U1U1 U2U2 diagonalise renormalisation group Therefore we can write: whereare some exponents some are positive (flow away from the critical surface) increase with iterations the others are negative (flow on the critical surface) decrease with iterations 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 y 1,y 2 k
RG
Linearisation:
11 22 Ejercicio 16 : k The critical line of the problem is given (linear approx.) by: