Preprint typeset in JHEP style - HYPER VERSION January 2016 The Quantum Hall Effect TIFR Infosys Lectures David Tong Department of Applied Mathematics and Theoretical Physics, Centre for Mathematical Sciences, Wilberforce Road, Cambridge, CB3 OBA, UK http://www.damtp.cam.ac.uk/user/tong/qhe.html
[email protected] { 1 { Recommended Books and Resources There are surprisingly few dedicated books on the quantum Hall effect. Two prominent ones are • Prange and Girvin, \The Quantum Hall Effect" This is a collection of articles by most of the main players circa 1990. The basics are described well but there's nothing about Chern-Simons theories or the importance of the edge modes. • J. K. Jain, \Composite Fermions" As the title suggests, this book focuses on the composite fermion approach as a lens through which to view all aspects of the quantum Hall effect. It has many good explanations but doesn't cover the more field theoretic aspects of the subject. There are also a number of good multi-purpose condensed matter textbooks which contain extensive descriptions of the quantum Hall effect. Two, in particular, stand out: • Eduardo Fradkin, Field Theories of Condensed Matter Physics • Xiao-Gang Wen, Quantum Field Theory of Many-Body Systems: From the Origin of Sound to an Origin of Light and Electrons Several excellent lecture notes covering the various topics discussed in these lec- tures are available on the web. Links can be found on the course webpage: http://www.damtp.cam.ac.uk/user/tong/qhe.html. Contents 1. The Basics 5 1.1 Introduction 5 1.2 The Classical Hall Effect 6 1.2.1 Classical Motion in a Magnetic Field 6 1.2.2 The Drude Model 7 1.3 Quantum Hall Effects 10 1.3.1 Integer Quantum Hall Effect 11 1.3.2 Fractional Quantum Hall Effect 13 1.4 Landau Levels 14 1.4.1 Landau Gauge 18 1.4.2 Turning on an Electric Field 21 1.4.3 Symmetric Gauge 22 1.5 Berry Phase 27 1.5.1 Abelian Berry Phase and Berry Connection 28 1.5.2 An Example: A Spin in a Magnetic Field 32 1.5.3 Particles Moving Around a Flux Tube 35 1.5.4 Non-Abelian Berry Connection 38 2.