Elementary Theory: the Incompressible Quantum Fluid

Abstract

In this lecture, I shall outline what I believe to be the correct fundamental picture of the fractional quantum Hall effect. The principal features of this picture are that the 1/3 state and its daughters are a new type of many-body condensate, that there are analogs of electrons and holes in the integral quantum Hall effect which in this case carry fractional charge, that the Hall plateau observed in experiments is due to localization of these fractionally charged quasiparticles, and that the rules for combining quasiparticles to make daughter states and excitons are simple. The key facts the theory has to explain are these:

• 1. The effect occurs when electrons are at a particular density, determined by the magnetic field strength. The separation between neighboring electrons locks in at particular values.

• 2. The fractional quantum Hall effect looks to the eye like the integral quantum Hall effect, except that the Hall conductance, in the case of the 1/3 step, is 1/3 e²h. There is a plateau. Changing the electron density a small amount does not alter the Hall conductance, but changing it a large amount does.

• 3. The 1/3 effect turns on in GaAs at about 1 K when the magnetic field is 15 Tesla.

• 4. The effect occurs only in the cleanest samples. Excessive dirt destroys it.

• 5. Other fractions (2/5, 2/7, …) also occur but they are more easily destroyed by dirt and require lower temperatures to be observed. The most ‘stable’ states have small denominators.