Introduction to the Fractional Quantum Hall Effect

Abstract

The quantum Hall effect (QHE) is one of the most remarkable condensed-matter phenomena dis-covered in the second half of the 20th century. It rivals superconductivity in its fundamental significance as a manifestation of quantum mechanics on macroscopic scales. The basic experimen-tal observation for a two-dimensional electron gas subjected to a strong magnetic field is nearly vanishing dissipation σ xx → 0 (1) and special values of the Hall conductance σ xy = ν e 2 h (2) given by the quantum of electrical conductance (e 2 /h) multiplied by a quantum number ν. This quantization is universal and independent of all microscopic details such as the type of semicon-ductor material, the purity of the sample, the precise value of the magnetic field, and so forth. As a result, the effect is now used to maintain (but not define) the standard of electrical resistance by metrology laboratories around the world. In addition, since the speed of light is now defined, a measurement of e 2 /h is equivalent to a measurement of the fine structure constant of fundamental importance in quantum electrodynamics. Fig. (1) shows the remarkable transport data for a real device in the quantum Hall regime. Instead of a Hall resistivity which is simply a linear function of magnetic field, we see a series of so-called Hall plateaus in which ρ xy is a universal constant ρ xy = − 1 ν h e 2 (3) independent of all microscopic details (including the precise value of the magnetic field). Associated with each of these plateaus is a dramatic decrease in the dissipative resistivity ρ xx −→ 0 which drops as much as 13 orders of magnitude in the plateau regions. Clearly the system is undergoing some sort of sequence of phase transitions into highly idealized dissipationless states. Just as in a superconductor, the dissipationless state supports persistent currents. In the so-called integer quantum Hall effect (IQHE) discovered by von Klitzing in 1980, the quantum number ν is a simple integer with a precision of about 10 −10 and an absolute accuracy of about 10 −8 (both being limited by our ability to do resistance metrology). In 1982, Tsui, Störmer and Gossard discovered that in certain devices with reduced (but still non-zero) disorder, the quantum number ν could take on rational fractional values. This so-called fractional quantum Hall effect (FQHE) is the result of quite different underlying physics involv-ing strong Coulomb interactions and correlations among the electrons. The particles condense into special quantum states whose excitations have the bizarre property of being described by fractional quantum numbers, including fractional charge and fractional statistics that are intermediate be-tween ordinary Bose and Fermi statistics. The FQHE has proven to be a rich and surprising arena for the testing of our understanding of strongly correlated quantum systems. With a simple twist of a dial on her apparatus, the quantum Hall experimentalist can cause the electrons to condense 54 Steven M. Girvin Séminaire Poincaré Figure 1: Integer and fractional quantum Hall transport data showing the plateau regions in the Hall resistance R H and associated dips in the dissipative resistance R. The numbers indicate the Landau level filling factors at which various features occur. After ref. [1]. into a bewildering array of new 'vacua', each of which is described by a different quantum field theory. The novel order parameters describing each of these phases are completely unprecedented. A number of general reviews exist which the reader may be interested in consulting [2–10] The present lecture notes are based on the author's Les Houches Lectures. [11] 2 Fractional QHE The free particle Hamiltonian an electron moving in a disorder-free two dimensional plane in a perpendicular magnetic field is H = 1 2m Π 2 (4) where Π ≡ p + e c A(r) (5) is the (mechanical) momentum. The magnetic field quenches the kinetic energy into discrete, massively degenerate Landau levels. In a sample of area L x L y , each Landau level has degeneracy equal to the number of flux quanta penetrating the sample N Φ = L x L y B Φ 0 = L x L y 2π 2 (6) where is the magnetic length defined by 1 2π 2 = B Φ 0 (7) and Φ 0 = h e 2 is the quantum of flux. The quantum number ν in the quantized Hall coefficient turns out to be given by the Landau level filling factor ν = N N Φ .