Broken Symmetry and Order Parameters

Transcription

1 BYU PHYS 731 Statistical Mechanics Chapters 8 and 9: Sethna Professor Manuel Berrondo Broken Symmetry and Order Parameters Dierent phases: gases, liquids, solids superconductors superuids crystals w/dierent symmetries quasi-crystals states w/fraction quantum Hall eect(chern-simmons) liquid crystals: neuatic, cholesteric, smectic,... magnetic states electrets spin glasses(disordered) Four steps(mermin) Identify the broken symmetry Dene the appropriate order parameter Examine the elementary excitations Classify the topological defects 1

2 Broken Symmetry Snapshot of water Rotational symmetry SO (2) by an arbitrary angle (continuous variable) vs. or d discrete symmetry C 6V (lower) ice vs. = sphere SO (3) cube O h Same physical system w/ dierent symmetries represent dierent phases (in equilibrium) if but not i Counterexample: liquid and gas have same symmetry 2

3 Sethna gures 3

4 More gures Spontaneous symmetry breaking Domian in Ferromagnet 4

5 Examples of order parameters(not unique!) Physical eld changing from one phase to the other Ferromagnets unit sphere M (x) = M ˆM (x) }{{} direction Magnetized material one dominant direction Order parameter space: S 2 Neumatic liquids ˆn (x) ˆn (x) Hemisphere w/ opposite points identied 5

6 Solids(lattice) Broken translational symmetry = Deformation described by vector u up to a rigid grid displacement: u u + n a n a = n 1 aê 1 + n 2 aê 2 u = u (x) unit cell vectors square lattices periodic B.C. T 2 torus Superuids and superconductors macroscopic wave function ψ = ψ e iφ φ (x)- order parameter space = S (circle) crystal order parameter space 6

7 Broken symmetry 1.- Parity in QM H = 1 2 P X2 odd even [H, P] =0 Pψ = ±ψ ψ even or odd T = X αx2 No degeneracy for ground state (i.e. tunneling) 2.- Parity in QFT Φ (x) eld H [Φ] = Φ 4 (x) αφ2 No tunneling # dimensions = the solution does not have the same symmetry as H [H, P] =0 but Φ (x) in neither odd nor even order = The symmetry is broken 7

8 3.- Continuous symmetries broken M (x) - magnetization eld symmetry group: SO (3) ferromagnets preferred direction symmetry group SO (2) symmetry is broken Elementary excitations lattice a translation by no energy cost low frequency displacements will cost little energy u( x) sound waves [ ε u (x), du ] ε dx [ ] du dx (1) Expansion: ε [u ] = 1 2 κ dx ( ) 2 du = 1 dx 2 κ dx (u ) 2 8

9 BYU PHYS 731 Statistical Mechanics Chapters 8 and 9: Sethna Professor Manuel Berrondo δε δu force density Sound waves: ρü = κ (u ) 2 u u x u u t Dispersion relation ρω 2 = κk 2 ω (k) = κ p k Goldstone theorem: FIGURE Φ (x) Φ (x) e iα When a const. symmetry is broken long wave length modulations in the symmetry direction have low frequency (Goldstone modes) solids low freq. phonons magnets spin waves nematic crystals rotational waves 9

10 Examples of Goldstone modes a) U (1) broken (as a global group) 2 massive elds 1 massive & 1 massless b) U (1) as a gauge group is broken(long range) Φ (x) Φ (x) e iα(x) + i α gauge 2 massive 1 photon 1 massive 1 massive photon! c) SO (3) broken(global symmetry) 3 massive scalar 1 massive 2 massless d) SO (3) broken as a gauge symmetry 3 massive scalar 3 massless vector 1 massive scalar 2 massive vector 1 massless vector = d. of f. b) Superconductivity Φ (x) Φ (x) e iα(x) A A 1 + ( ) c Λ 1 Φ = R i 2m φ φ e φ 2 A density Maxwell B = j A 2 B = k 2 B 10

11 Topological defects or Generalization of Cauchy's theorem Homotopy groups Magnets hedge hog defect but no line defects Nematics line defects two defects annhilate one another 11