1. Ginzburg-Landau theory of second-order phase transitions Vitaly L. Ginzburg Lev Landau Second order= no latent heat (ferromagnetism, superfluidity, superconductivity). Let the transition occur at T=T C Ordered phase below transition (typically) Order parameter  (vanishes in disordered phase):  = macrospopic wave function (superconductivity)  = magnetization (ferro or antiferro) 1 Order parameter  = excess population at angle  (or a simple function of  ) in nematic-isotropic transition in liquid crystals-simplest example (1 real scalar parameter) Alignment in a nematic phase. One of the most common LC phases is the nematic. The word nematic comes from the Greek νημα (nema ), which means "thread”Greek Example: liquid crystals

2. 2 (see lectures by A. Salonen) Tfy-0.3252 Soft Matter Physics, Fall 2009 / E. Salonen

3. 3 Gibbs free energy Gibbs free energy must be minimum at equilibrium at any T Equilibrium Condition Near the transition Gibbs free energy must be minimum at equilibrium in disorderesìd phase Therefore the expansion around critical point starts from second order At  =0 there is a minimum or a maximum Order parameter  is 0 or small (since it vanishes in disordered phase) How G depends on  near the transition Temperature

4. Above T C minimum at  =0 (order parameter vanishes at equilibrium in disordered phase Below T C maximum at  =0 4 G-G 0 GL Phenomenological model, with terms up to 4° order: At the transition T= T C G must be the same for nematic and isotropic (equilibrium).

5. One obtains the discontinuity of order parameter by imposing equilibrium and minimum at Tc 5 often written in the form Discontinuity of Order Parameter  at T c

6. 6 Above, we got the discontinuity Order parameter versus T To obtain  (T) we can start from the equilibrium condition One can write T* in terms of T C and the other parameters. The order parameter is discontinuous at transition. Insert  c back into G-G 0 equation T*

7. 7 To get rid of the double sign, choose the right h at the transition:

8. Langevin paramagnetism Paul Langevin (1872-1946) 8

9. 9 PIERRE-ERNEST WEISS born March 25, 1865, Mulhouse, France. died Oct. 24, 1940, Lyon, France. Weiss mean field theory (1907) of Ferromagnetism: One can fix the parameter in terms of T c as follows: The spontaneous magnetization may remain even for H=0

10. 1 solution if T>T c, 2 otherwise x=y

11. 11 PIERRE-ERNEST WEISS born March 25, 1865, Mulhouse, France. died Oct. 24, 1940, Lyon, France. T  Tc from below: Spontaneous magnetization for H=0 From the graphical solution we gain the trend with T of the M(T) curve. Increasing T at H=0 M must vanish, but how?

12. 12 A different experiment: fix T =Tc and measure M dependence on H at for H  0. Since M is small we can expand

13. Behavior for T > T c, in paramagnetic phase M small How do these critical exponents arise? Widom in 1965 put forth the scaling hypothesis for magnetic materials in field B: This produces relations between various critical exponents. The dependence on approximation scheme and dimensionality is dramatic.

14. 14 (Weiss theory)

15. Ising Model in 1d, defined on a closed ring where the sums over sites and s n is a classical spin taking the values +1,-1. The Partition function is defined by: 15 (model invented in 1920 by Wilhelm Lenz as a thesis for Ernst Ising)

16. 16 Rearrange: There is a [ ] factor for each bond; each factor is a matrix product 1 2

17. 17

18. 18 For H  0,

19. 19 We need to note that another reasoning is possible. Instead of This is transfer matrix has the same trace and the same determinant as the previous one, hence the eigenvalues are the same. one can decide to distribute the interactions with H symmetrically among the bonds and define a different transfer matrix, which is also common in textbooks, with elements: 12 12

20. 20 Thermodynamic limit: only large eigenvalue matters The peak in specific heat is due to some short-range order (increasing T at low temperatures spins are reversed and this costs energy).

21. 21 no ferromagnetism, no phase transition, no long-range order! 1d theory fails to explain magnetism. Why? no ferromagnetism, no phase transition, no long-range order! 1d theory fails to explain magnetism. Why? For large N there is a gain in F at any T. In 2d is different.

22. 22 Rudolf Peierls Centre for Theoretical Physics 1 Keble Road Oxford, OX1 3NP England

23. Percolation problem: Renormalization group approach to the onset of conductivity Consider a regular lattice (sites connected by bonds, with Z bonds per site) Remove a fraction 1-p of bonds at random. Conductivity is a function of p. There is a critical value p c such that for p p c the film as a whole is conducting, even if it contains nonconducting islands. The macroscopic conductivity  of the lattice is a function  (p,  bond conductivity. A.P.Young and R.B. Stinchcombe J. Phys. C 8 L535 (1975) How to find an approximation to p c ? By scaling! For exampe, discard half of the x and half of the y and consider a new lattice where the nodes can be connected or not with some probability. The new problem is similar. We can exploit that! How to find an approximation to p c ? By scaling! For exampe, discard half of the x and half of the y and consider a new lattice where the nodes can be connected or not with some probability. The new problem is similar. We can exploit that! 23 Let us consider a square lattice:

24. 24 Decimation

25. A similar problem can be posed on the lattice with vertices 1,2,.. or with the rescaled lattice A,B,…on a larger square mesh. p 1 = probability in rescaled network,    bond conductivity in rescaled network  (p 1,    25

26. By comparison with the exact results one can evaluate approximations.

27. Consider only paths involving  A 1B 2     C D What is the probability p 1 that nodes A and B of new lattice are connected? 27

28. 28