If you followed what we learned about the integer quantum Hall effect, you’ll remember that we used the pumping argument to establish that the Hall conductance in an incompressible liquid is quantized in integers. You might now wonder if the experimental evidence for the fractional quantum Hall effect completely invalidates this argument in some way. The key step in the argument was to realize that the pumped charge that results from the insertion of one flux quantum \(\Phi_0\) into a Corbino geometry is

\[ Q_{pump}=\sigma_{xy}\Phi_0\equiv \nu e, \]

where \(\nu=\sigma_{xy}/G_0\) is the Hall conductance in dimensionless units. For the non-interacting system that we studied in the quantum Hall effect, we assumed that only an integer number of electrons could be transferred between the edges - so \(\nu\) had to be an integer.

The real reason that the charge transferred had to be an integer multiple of the electron charge was that the Hamiltonian for the electrons was identical between flux \(\Phi=0\) and flux \(\Phi=\Phi_0\). Since the system is incompressible, it is reasonable to assume that all excitations in the system are local. Usually we expect different excited states to differ by rearranging electrons. Within this framework, such excitations can differ by integer multiples of electronic charge.

The existence of fractional values of \(\nu\) implies that the edge can have local excitations that differ by a fractional electron charge. In principle, the inner edge of the Corbino geometry can be shrunk to a point, and if we do this, we’re forced to conclude that the system can now host excitations that have fractional charge.

The fractional charged excitations are local particles just like the electrons themselves. So we can ask about the statistics under exchange of two such particles. On performing such an exchange, the total many-body wave function of the system returns to itself, but the wave function can pick up a Berry phase. For fermions this phase is \(\pi\) and for Bosons it is zero. Instead of computing the phase directly, let us consider doing a double exchange, which is topologically equivalent to taking a particle around another one and computing the phase for that.

Let us first assume that one of the particles was created by a flux quantum. Since the flux quantum created this particle adiabatically by a pumping process, locality dictates that the particle going around the flux quantum + particle cannot know about the existence of the other particle. Thus the phase from going around a particle together with its flux quantum must vanish. On the other hand, the particle picks up a phase of \(2\pi \nu\) from just going around the flux quantum. Thus double exchange of a pair of particles leads to a Berry phase of \(-2\pi\nu\). This is another strange property of excitations in the FQH state! They must obey different statistics than both fermions and bosons, and are thus referred to as anyons. Therefore the exchange phase of anyons in the simple FQH states is given by

\[ \phi_{exch}=\pi\nu. \]