Order Parameters, Broken Symmetry, and TopologyJim Sethna, http://www.lassp.cornell.edu/sethna/OrderParameters/Four Elements of Ancient Greeks: Earth, Water, Air, FireWe have phases: solid, liquid, gas, plasma?Crystals (200), quasicrystals, magnets, metals, insulators, superconductors, superfluids, liquid crystals (nematic, helical, blue phase I, II, and fog, smectic A, B, C, C*, D, I, …), quantum Hall, spin glasses, electroweak (four kinds of photons, now ν, W+, W-, ZQuasicrystal: icosahedral symmetryThe System• Too many cases: need systematic approach• Often, new phases don’t fit into old system(quasicrystals, fractional quantum Hall, spin glasses…) 1. Identify Broken Symmetry2. Pick Order Parameter Field3. Examine Elementary Excitations4. Classify Topological Defects1. Identify the Broken SymmetryRotational Symmetry Broken to CubicEnsemble Average:Rotational Symmetry Broken to Hexagonal,Translational Symmetry Broken to Lattice TranslationsLiquid Crystal: Rotation (Translation)Magnet: Rotational, Time ReversalSuperfluids: Gauge SymmetryCrystalsOften can tell if same symmetry by adiabatic continuity: oil to alcohol to water 2. Define the Order ParameterMagnetsMagnet• Order parameter M(x), local average magnetization• Often spatially varying (unmagnetized iron many domains in different directions)• Roughly fixed in magnitude: • Sphere S2Order Parameter• Summarizes ``important info’’ about local state of material• Coarse-grains away most degrees of freedom (M ignoresatoms, electrons)• Order parameter as field (arrow at each point)• Order parameter as mapping (M(x) takes R3into S2)IsingOP SpaceS0M(x)2. Define the Order ParameterNematic Liquid Crystal• Long Thin Molecules• Old LCD displays (watches)• Long axes align• Head-to-tail disorder• Ground states labeled by headless arrow n≡-n• Order Parameter Space Hemisphere• Antipodal Points on Equator Aligned• RP22. Define the Order ParameterTwo-Dimensional CrystalSquare lattice, spacing au(x) aligns ideal lattice onto local deformed latticeAmbiguity: u≡u+(m a, n a)A square with periodic boundary conditions is a torus(doughnut, bagel, inner tube)Order parameter space