Fractional charge and anyonic statistics

For a Laughlin state at filling one over m, inserting the appropriate flux creates a quasihole with charge positive e divided by m relative to the electron liquid. Its density deficit is spread over a microscopic correlation scale rather than attached to a smaller fundamental electron. A localized quasihole wavefunction can be constructed by multiplying by one factor connecting each electron coordinate to the quasihole position. Charge determination requires the liquid's screening and adiabatic-response assumptions, not merely counting a polynomial zero.

Several quasiholes combine into Abelian sectors labelled by their number modulo m after quotienting by local electron excitations. The charge of a sector is correspondingly specified modulo an integer electron charge, with any attached local particle recorded separately. Such sector labels are nonlocal properties of a gapped phase. They cannot be distinguished by arbitrary local rearrangements far from the excitation, but a physical detector still sees ordinary nonnegative probabilities.

For elementary identical Laughlin quasiholes, a chosen oriented exchange contributes pi over m, whereas moving one all the way around another contributes twice that angle. Confusing these two paths produces a factor-of-two error. Reversing an oriented braid reverses its Abelian statistical phase. The overall phase is only defined modulo two pi, and comparing it experimentally requires interfering suitable histories rather than reading an absolute phase of one isolated ket.

A moving charged quasihole also accumulates an Aharonov–Bohm phase from enclosed magnetic flux, and energy integrated over the path produces a dynamical phase. Changing a path's area or duration can therefore change its total phase even without changing the braid's topology. Extracting statistics requires a protocol that controls, cancels, or calibrates those additional terms. A total phase equal to zero modulo two pi does not by itself establish bosonic exchange.

Abelian sectors fuse into a unique sector for a specified pair of labels, so well separated braids act by phases. Non-Abelian anyons can have more than one allowed fusion channel. Several excitations with fixed total charge then support a state space on which braids act by matrices. Different braid sequences can fail to commute even after common dynamical phases are removed. This is stronger structure than assigning one exotic exchange angle to every trajectory.

For example, the Ising fusion rule sigma times sigma allows either the vacuum or a fermion channel. A measurement of that channel is different from simply counting the number of excitations. Reliable topological manipulation also requires a gap, large separation compared with correlation length, slow motion relative to bulk excitations, and control of residual splitting and poisoning. Fractional statistics is established phase structure; claims of complete protection or a particular experimental realization need separate evidence and error assumptions.