Quasiparticles do the twist

One of the most memorable lessons of an undergraduate course in quantum mechanics is that identical particles can have two types of “statistics.” An exchange of two identical bosons leaves the many-particle wave function unchanged, while an exchange of two identical fermions introduces a minus sign. The consequences of this minus sign are profound in all areas of physics. Moreover, it has long been known that twodimensional systems can have statistics that are neither bosonic nor fermionic. Robert Laughlin’s explanation of the fractional quantum Hall effect introduced quasiparticles with fractional charges such as e/3. These are thought to possess fractional statistics; if you exchange two identical quasiparticles, the wave function picks up a factor exp(iφ). In more complex quantum Hall states, different quasiparticle exchanges, represented by matrices, do not always commute: the quasiparticles obey non-Abelian statistics.