"local" Landau-like term

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

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

8. Ginzburg criterion Levanyuk criterion

11. d: dimensionality

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

27. ’ ’ (H’, T’)

28. majority rule x x

29. close to the critical point! homogeneity property

30. dimensionality EJERCICIO 15

32. Renormalisation group transformation H=0 In practice, this only works near the critical point. At the critical point x does not change on RG transformation! The renormalisation group exploits properties at and near T=Tc

34. RG

35. the K0 parameter is needed! but only K1 and K2 are relevant

37. x=0 x= 8

38. x=0 x= 8 x=0 T=0 or T= attractive 8 repulsive mixed NON-TRIVIAL FIXED POINT

39. to trivial fixed point T=0 (point with x= 0) (points with x= ) to trivial fixed point T= 8 8 (point with x= 0)

40. k =K’=K*+k’

41. diagonalise … k renormalisation group SCALING FIELDS Therefore we can write: U1 U2 where are some exponents some are positive (flow away from the critical surface) la increase with iterations the others are negative (flow on the critical surface) la 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 y1,y2

42. RG

43. 1.721 -0.387

44. Linearisation:

45. The critical line of the problem is given (linear approx.) by: k f2 f1 Ejercicio 16: