1. "local" Landau-like term gradient term taking care of fluctuations free energy of disordered phase ("effective Hamiltonian") cost involved in creating inhomogeneities

2. average value of order parameter  = 0 in disordered phase  = 0 in ordered phase contribution from fluctuations

8. Ginzburg criterion Levanyuk criterion

11. d: dimensionality

25. T=T c : fractal structure fluctuations of all length scales possible no typical length scale

27. ’ ’ (H’, T’)

28.   majority rule

29. homogeneity property close to the critical point!

30. dimensionality EJERCICIO 15

32. 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

34. RG

35. the K 0 parameter is needed! but only K 1 and K 2 are relevant

37.  =0  = 

38. repulsive attractive mixed  =0 T=0 or T=  NON-TRIVIAL FIXED POINT  =0  = 

39. to trivial fixed point T=0 to trivial fixed point T=  (points with  = )  (point with  = 0) (point with  = 0)

40. k =K’=K*+k’

41. 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

42. RG

43. 1.721-0.387

44. Linearisation:

45. 11 22 Ejercicio 16 : k The critical line of the problem is given (linear approx.) by: