Aspects topologiques de la physique en basse dimension. Topological aspects of low dimensional systems

Transcription

1 Les Houches - Ecole d'ete de Physique Theorique 69 Aspects topologiques de la physique en basse dimension. Topological aspects of low dimensional systems Topological aspects of low dimensional systems Bearbeitet von A Comtet, T Jolicoeur, S Ouvry, F David 1. Auflage Buch. XXXIV, 911 S. Hardcover ISBN Format (B x L): 15,5 x 23,5 cm Gewicht: 1580 g Weitere Fachgebiete > Physik, Astronomie > Physik Allgemein > Theoretische Physik, Mathematische Physik Zu Leseprobe schnell und portofrei erhältlich bei Die Online-Fachbuchhandlung beck-shop.de ist spezialisiert auf Fachbücher, insbesondere Recht, Steuern und Wirtschaft. Im Sortiment finden Sie alle Medien (Bücher, Zeitschriften, CDs, ebooks, etc.) aller Verlage. Ergänzt wird das Programm durch Services wie Neuerscheinungsdienst oder Zusammenstellungen von Büchern zu Sonderpreisen. Der Shop führt mehr als 8 Millionen Produkte.

2 CONTENTS Lecturers Participants Préface Preface Contents xi xiii xvii xxi xxiii Course 1. Electrons in a Flatland by M. Shayegan 1 1 Introduction 3 2 Samples and measurements DelectronsattheGaAs/AlGaAsinterface Magnetotransportmeasurementtechniques Ground states of the 2D System in a strong magnetic field Shubnikov-de Haas oscillations and the IQHE FQHEandWignercrystal Composite fermions 16 5 Ferromagnetic state at ν = 1 and Skyrmions 19 6 Correlated bilayer electron states Overview ElectronSysteminawide,single,quantumwell EvolutionoftheQHEstatesinawidequantumwell Evolutionofinsulatingphases Many-body, bilayer QHE at ν = SpontaneousinterlayerChargetransfer Summary... 48

3 xxiv Course 2. The Quantum Hall Effect: Novel Excitations and Broken Symmetries by S.M.Girvin 53 1 The quantum Hall effect Introduction Why2Disimportant Constructingthe2DEG Whyisdisorderandlocalizationimportant? Classicaldynamics Semi-classicalapproximation QuantumdynamicsinstrongBFields IQHEedgestates Semiclassicalpercolationpicture FractionalQHE The ν =1many-bodystate Neutralcollectiveexcitations Chargedexcitations FQHEedgestates Quantumhallferromagnets Coulombexchange Spinwaveexcitations Effectiveaction Topologicalexcitations Skyrmiondynamics Skyrmelattices Double-layerquantumHallferromagnets Pseudospinanalogy Experimentalbackground Interlayerphasecoherence Interlayer tunneling and tilted field effects Appendix A Lowest Landau level projection 165 Appendix B Berry s phase and adiabatic transport 168 Course 3. Aspects of Chern-Simons Theory by G.V. Dunne Introduction Basics of planar field theory Chern-Simonscoupledtomatterfields- anyons Maxwell-Chern-Simons: Topologically massive gauge theory Fermionsin2+1-dimensions Discrete symmetries: P, C and T

4 xxv 2.5 Poincaré algebrain2+1-dimensions NonabelianChern-Simonstheories Canonical quantization of Chern-Simons theories CanonicalstructureofChern-Simonstheories Chern-Simonsquantummechanics Canonical quantization of abelian Chern-Simons theories Quantizationonthetorusandmagnetictranslations Canonical quantization of nonabelian Chern-Simons theories Chern-Simons theories with boundary Chern-Simons vortices Abelian-Higgs model and Abrikosov-Nielsen-Olesen vortices RelativisticChern-Simonsvortices NonabelianrelativisticChern-Simonsvortices NonrelativisticChern-Simonsvortices:Jackiw-Pimodel NonabeliannonrelativisticChern-Simonsvortices Vortices in the Zhang-Hansson-Kivelson model for FQHE Vortexdynamics Induced Chern-Simons terms Perturbatively induced Chern-Simons terms: Fermion loop Induced currents and Chern-Simons terms Induced Chern-Simons terms without fermions Afinitetemperaturepuzzle Quantummechanicalfinitetemperaturemodel Exactfinitetemperature2+1effectiveactions Finite temperature perturbation theory and Chern-Simons terms. 256 Course 4. Anyons by J. Myrheim Introduction Theconceptofparticlestatistics Statisticalmechanicsandthemany-bodyproblem Experimentalphysicsintwodimensions Thealgebraicapproach:Heisenbergquantization Moregeneralquantizations The configuration space TheEuclideanrelativespacefortwoparticles Dimensions d = 1,2, Homotopy Thebraidgroup Schrödinger quantization in one dimension 286

5 xxvi 4 Heisenberg quantization in one dimension Thecoordinaterepresentation Schrödinger quantization in dimension d Scalarwavefunctions Homotopy Interchangephases Thestatisticsvectorpotential The N-particle case Chern-Simonstheory The Feynman path integral for anyons EigenstatesforPositionandmomentum Thepathintegral Conjugation classes in S N Thenon-interactingcase Duality of Feynman and Schrödinger quantization The harmonic oscillator The two-dimensional harmonic oscillator Two anyons in a harmonic oscillator potential Morethantwoanyons Thethree-anyonproblem The anyon gas Theclusterandvirialexpansions Firstandsecondorderperturbativeresults Regularization by periodic boundary conditions Regularization by a harmonic oscillator potential Bosonsandfermions Twoanyons Threeanyons TheMonteCarlomethod The path integral representation of the coefficients G P Exactandapproximatepolynomials Thefourthvirialcoefficientofanyons Twopolynomialtheorems Charged particles in a constant magnetic field Oneparticleinamagneticfield Twoanyonsinamagneticfield Theanyongasinamagneticfield...380

6 xxvii 10 Interchange phases and geometric phases Introductiontogeometricphases Oneparticleinamagneticfield Twoparticlesinamagneticfield Interchangeoftwoanyonsinpotentialwells Laughlin stheoryofthefractionalquantumhalleffect Course 5. Generalized Statistics in One Dimension by A.P. Polychronakos Introduction Permutation group approach RealizationofthereducedHilbertspace Pathintegralandgeneralizedstatistics Cluster decomposition and factorizability One-dimensional systems: Calogero model TheCalogero-Sutherland-Mosermodel Large-NpropertiesoftheCSMmodelandduality One-dimensional systems: Matrix model Hermitianmatrixmodel Theunitarymatrixmodel Quantizationandspectrum Reductiontospin-particlesystems Operator approaches Exchangeoperatorformalism Systemswithinternaldegreesoffreedom AsymptoticBetheansatzapproach Thefreezingtrickandspinmodels Exclusion statistics MotivationfromtheCSMmodel Semiclassics Heuristics Exclusionstatisticalmechanics Exclusionstatisticspathintegral Isthistheonly exclusion statistics? Epilogue 469

7 xxviii Course 6. Lectures on Non-perturbative Field Theory and Quantum Impurity Problems by H. Saleur Some notions of conformal field theory The free boson via pathintegrals NormalorderingandOPE Thestressenergytensor Conformalin(co)variance SomeremarksonWardidentitiesinQFT TheVirasoroalgebra:Intuitiveintroduction Cylinders ThefreebosonviaHamiltonians Modularinvariance Conformal invariance analysis of quantum impurity fixed points Boundary conformal field theory Partition functions and boundary states Boundary entropy The boundary sine-gordon model: General results Themodelandtheflow PerturbationneartheUVfixedpoint PerturbationneartheIRfixedpoint An alternative to the instanton expansion: The conformal invarianceanalysis Search for integrability: Classical analysis Quantum integrability Conformalperturbationtheory S-matrices Back to the boundary sine-gordon model The thermodynamic Bethe-ansatz: The gas of particles with Yang-Baxter statistics ZamolodchikovFateevalgebra TheTBA AStandardcomputation:ThecentralCharge Thermodynamics of the flow between N and D fixed points Using the TBA to compute static transport properties Tunneling in the FQHE Conductance without impurity Conductance with impurity

8 xxix Seminar 1. Quantum Partition Noise and the Detection of Fractionally Charged Laughlin Quasiparticles by D.C. Glattli Introduction Partition noise in quantum conductors Quantumpartitionnoise Partitionnoiseandquantumstatistics Quantum conductors reach the partition noise limit Experimental evidences of quantum partition noise in quantum conductors Partition noise in the quantum Hall regime and determination of the fractional Charge EdgestatesintheintegerquantumHalleffectregime Tunneling between IQHE edge channels and partition noise Edgechannelsinthefractionalregime Noisepredictionsinthefractionalregime MeasurementofthefractionalChargeusingnoise BeyondthePoissoniannoiseoffractionalcharges Course 7. Mott Insulators, Spin Liquids and Quantum Disordered Superconductivity by Matthew P.A. Fisher Introduction Models and metals Noninteractingelectrons Interactioneffects Mott insulators and quantum magnetism Spinmodelsandquantummagnetism Spinliquids Bosonization primer Leg Hubbard ladder Bondingandantibondingbands Interactions Bosonization d-mottphase Symmetryanddoping...603

9 xxx 6 d-wave superconductivity BGStheoryre-visited d-wavesymmetry Continuumdescriptionofgaplessquasiparticles Effective field theory Quasiparticlesandphaseflucutations Nodons Vortices ic/2e versus hc/e vortices Duality Nodal liquid phase Half-filling Dopingthenodalliquid Closingremarks Appendix A Lattice duality 635 A.1 Twodimensions A.2 Threedimensions Course 8. Statistics of Knots and Entangled Random Walks by S. Nechaev Introduction Knot diagrams as disordered Spin Systems Briefreviewofstatisticalproblemsintopology Abelian problems in statistics of entangled random walks and incompletenessofgaussinvariant Nonabelianalgebraicknotinvariants Lattice knot diagrams as disordered Potts model Notion about annealed and quenched realizations of topological disorder Random walks on locally non-commutative groups Brownian bridges on simplest non-commutative groups and knot statistics Randomwalksonlocallyfreegroups Analytic results for random walks on locally free groups Brownian bridges on Lobachevskii plane and products ofnon-commutativerandommatrices...697

10 xxxi 4 Conformal methods in statistics of random walks with topological constraints Construction of nonabelian connections for Γ 2 and PSL(2, Z) from conformalmethods Random walk on double punctured plane and conformal field theory Statistics of random walks with topological constraints in the two dimensionallatticesofobstacles Physical applications. Polymer language in statistics of entangled chain-like objects Polymerchainin3D-arrayofobstacles Collapsedphaseofunknottedpolymer Some tight problems of the probability theory and statistical physics RemarksandcommentstoSection RemarksandcommentstoSections3and RemarksandcommentstoSection Seminar 2. Twisting a Single DNA Molecule: Experiments and Models by T. Strick, J.-F. Allemand, D. Bensimon, V. Croquette, C. Bouchiat, M. Mézard and R. Lavery Introduction Single molecule micromanipulation Forcesatthemolecularscale Brownianmotion:Asensitivetoolformeasuringforces Stretching B-DNA is well described by the worm-like chain model TheFreelyjointedchainelasticitymodel Theoverstretchingtransition The torsional buckling instability The buckling instability at T = The buckling instability in the rod-like chain (RLC) model ElasticrodmodelofsupercoiledDNA Theoretical analysis of the extension versus supercoiling experiments Criticaltorquesareassociatedtophasechanges...754

11 xxxii 5 Unwinding DNA leads to denaturation Twisting rigidity measured through the critical torque ofdenaturation Phase coexistence in the large torsional stress regime Overtwisting DNA leads to P-DNA Phase coexistence of B-DNA and P-DNA in the large torsional stressregime Chemicalevidenceofexposedbases Conclusions 762 Course 9. Introduction to Topological Quantum Numbers by D.J. Thouless 767 Preface Winding numbers and topological classification Precisionandtopologicalinvariants Windingnumbersandlinedefects Homotopygroupsanddefectclassification Superfluids and superconductors Quantizedvorticesandfluxlines Detectionofquantizedcirculationandflux Precision of circulation and flux quantization measurements The Magnus force Magnusforceandtwo-fluidmodel Vortexmovinginaneutralsuperfluid Transverse force in superconductors Quantum Hall effect Introduction Proportionality of current density and electric field Bloch stheoremandthelaughlinargument Chernnumbers FractionalquantumHalleffect Skyrmions Topological phase transitions The vortex induced transition in superfluid helium films Two-dimensionalmagneticSystems Topologicalorderinsolids Superconducting films and layered materials

12 xxxiii 6 The A phase of superfluid 3 He Vortices in the A phase Otherdefectsandtextures Liquid crystals Orderinliquidcrystals Defectsandtextures Seminar 3. Geometrical Description of Vortices in Ginzburg-Landau Billiards by E. Akkermans and K. Mallick Introduction Differentiable manifolds Manifolds Differentialformsandtheirintegration Topologicalinvariantsofamanifold Riemannianmanifoldsandabsolutedifferentialcalculus TheLaplacian Bibliography Fiber bundles and their topology Introduction Localsymmetries.Connexionandcurvature Chernclasses Manifolds with a boundary: Chern-Simons classes The Weitzenböckformula The dual point of Ginzburg-Landau equations for an infinite System TheGinzburg-Landauequations The Bogomol nyi identities The superconducting billiard Thezerocurrentline Aselectionmechanismandtopologicalphasetransitions A geometrical expression of the Gibbs potential for finite Systems. 874 Seminar 4. The Integer Quantum Hall Effect and Anderson Localisation by J.T. Chalker Introduction Scaling theory and localisation transitions 882

13 xxxiv 3 The plateau transitions as quantum critical points Single particle models Numerical studies Discussion and outlook 892 Seminar 5. Random Magnetic Impurities and Quantum Hall Effect by J. Desbois Average density of states (D.O.S.) [1] Hall conductivity [2] Magnetization and persistent currents [3] 904 Seminars by participants 911