Quantum Hall Effects

1 Preface The present notes cover a series of three lectures on the quantum Hall effect given at the Singapore session " Ultracold Gases and Quantum Information " at Les Houches Summer School 2009. Almost 30 years after the discovery of the quantum Hall effect, the research subject of quantum Hall physics has certainly acquired a certain degree of maturity that is reflected by a certain number of excellent reviews and books, of which we can cite only a few (Prange and Girvin, 1990; Yoshioka, 2002; Ezawa, 2000) for possible further or complementary reading. Also the different sessions of Les Houches Summer School have covered in several aspects quantum Hall physics, and S. M. Girvin's series of lectures in 1998 have certainly become a reference in the field. Girvin's lecture notes were indeed extremely useful for me myself when I started to study the quantum Hall effect at the beginning of my Master and PhD studies. The present lecture notes are complementary to the existing literature in several aspects. One should first mention its introductory character to the field, which is in no way exhaustive. As a consequence, the presentation of one-particle physics and a detailed discussion of the integer quantum Hall effect occupy the major part of these lecture notes, whereas the – certainly more interesting – fractional quantum Hall effect, with its relation to strongly-correlated electrons, its fractionally charged quasi-particles and fractional statistics, is only briefly introduced. Furthermore, we have tried to avoid as much as possible the formal aspects of the fractional quantum Hall effect, which is discussed only in the framework of trial wave functionsà la Laughlin. We have thus omitted, e.g., a presentation of Chern-Simons theories and related quantum-field theoretical approaches, such as the Hamiltonian theory of the fractional quantum Hall effect (Murthy and Shankar, 2003), as much as the relation between the quantum Hall effect and conformal field theories. Although these theories are extremely fruitful and still promising for a deeper understanding of quantum Hall physics, a detailed discussion of them would require more space than these lecture notes with their introductory character can provide. Another complementary aspect of the present lecture notes as compared to existing textbooks consists of an introduction to Landau-level quantisation that treats in a parallel manner the usual non-relativistic electrons in semiconductor heterostructures and relativistic electrons in graphene (two-dimensional graphite). Indeed, the 2005 discovery of a quantum Hall effect …