The Quantum Hall Effect

1 Introduction.- 1.1. Introduction.- 1.2. Preview of Coming Attractions.- 1.3. The Ordinary Hall Effect.- 1.4. Measuring the Conductance.- 1.5. Introduction to the Quantum Case.- 1.6. Impurity Effects.- 1.7. Gauge Arguments.- 1.8. Inversion Layers.- 1.9. Acknowledgements.- 1.10. Notes.- 1.11. Problems.- A The Integer Effect.- 2 Experimental Aspects and Metrological Applications.- 2.1. Basic Principles.- 2.2. The Devices.- 2.3. The Basic Experiment.- 2.4. Initial Experiments.- 2.5. Precision Measurement Techniques.- 2.6. Quantum Hall Resistors.- 2.7. An Absolute Resistance Standard.- 2.8. The Fine-Structure Constant.- 2.9. Temperature Dependence of ?xx.- 2.10. Temperature Dependence of ?xy.- 2.11. Current Distribution and Edge Effects.- 2.12. Current Dependence and Breakdown.- 2.13. Hall Step Widths and Shapes.- 2.14. Thermomagnetic Transport.- 2.15. Magnetic Moment.- 2.16. Magnetocapacitance.- 2.17. Magneto-Photoresponse.- 2.18. Conclusions.- 2.19. Acknowledgements.- 2.20. Notes.- 3 Effects of Imperfections and Disorder.- 3.1. Introduction.- 3.2. The Impurity Potential.- 3.3. Weak Potential.- 3.4. Scattering Potential.- 3.5. Smooth Potential.- 3.6. Continuum Percolation Model.- 3.7. General Potential.- 3.8. Numerical Studies.- 3.9. Acknowledgements.- 3.10. Notes.- 3.11. Problems.- 4 Topological Considerations.- 4.1. Topological Quantum Numbers.- 4.2. Quantum Hall Effect in a Periodic Potential.- 4.3. Generalization of the Topological Interpretation.- 4.4. Notes.- 5 Field Theory, Scaling and the Localization Problem.- 5.1. Introduction.- 5.2. Basic Notions.- 5.3. The Search for the Principle.- 5.4. Structure of the Effective Field Theory.- 5.5. Instantons and Scaling.- 5.6. Notes.- B: The Fractional Effect.- 6 Experimental Aspects.- 6.1. Introduction.- 6.2. Conditions for the Observation of the FQHE and the Choice of Semiconductor Systems.- 6.3. The Fractional Quantum Hall Effect.- 6.4. Magneto-Transport in Two Dimensions.- 6.5. Future Experiments.- 6.6. Acknowledgements.- 6.7. Notes.- 7 Elementary Theory: The Incompressible Quantum Fluid.- 7.1. Introduction.- 7.2. Quantized Motion of Small Numbers of Electrons.- 7.3. Variational Ground State.- 7.4. Computational Methods.- 7.5. Fractionally Charged Quasiparticles.- 7.6. Exactness of the Fractional Quantum Hall Effect.- 7.7. More Than One Quasiparticle.- 7.8. Conclusions.- 7.9. Acknowledgements.- 8 The Hierarchy of Fractional States and Numerical Studies.- 8.1. Introduction.- 8.2. Pseudopotential Description of Interacting Particles with the Same Landau Index.- 8.3. The Principle Incompressible States: A Non-Variational Derivation.- 8.4. Quasiparticle and Quasihole Excitations.- 8.5. The Hierarchy of Quasiparticle—Quasihole Fluids.- 8.6. Translationally Invariant Geometries for Numerical Studies.- 8.7. Ground-State Energy Studies.- 8.8. Pair Correlations and the Structure Factors.- 8.9. Quasiparticle Excitations.- 8.10. Collective Excitations.- 8.11. Acknowledgements.- 9 Collective Excitations.- 9.1. Introduction.- 9.2. Density Waves as Elementary Excitations.- 9.3. Collective Modes in the FQHE.- 9.4. Magnetophonons and Magnetorotons.- 9.5. Further Superfluid Analogies: Vortices.- 9.6. Quasi-Excitons.- 9.7. Magnetoplasmons and Cyclotron Resonance.- 9.8. Other Collective Modes.- 9.9. Role of Disorder.- 9.10. Acknowledgements.- 9.11. Notes.- 9.12. Problems.- C: The Quantum Hall Effect.- 10 Summary, Omissions and Unanswered Questions.- 10.1. Integer Effect at Zero Temperature.- 10.2. IQHE at Finite Temperatures.- 10.3. Metrology.- 10.4. Basic Picture of the Fractional Effect.- 10.5. Remarks on the Neglect of Landau Level Mixing.- 10.6. Wanted: More Experiments.- 10.7. Towards a Landau—Ginsburg Theory of the FQHE.- 10.8. Acknowledgements.- 10.9. Notes.- 10.10. Problems.- Appendix Recent Developments.- A.1. Introduction.- A.2. Off-Diagonal Long-Range Order.- A.3. Ring Exchange Theories.- A.5. Even-Denominator and Spin-Reversed States.- A.6. Spin-Reversed Hierarchy.- A.7. Summary and Conclusions.- References.