Anyon

An anyon is a two-dimensional quasiparticle whose exchange or braiding statistics can differ from bosonic and fermionic statistics.

The topological spin ${\theta }_a$ is the phase produced by a $2\pi$ rotation of anyon $a$, equivalently its self-exchange phase in an Abelian anyon theory. The two-dimensional spin-statistics relation is ${\theta }_a=e^{2\pi i{s}_a}$.

A charge-flux composite carries electric charge $q$ and localized magnetic flux $\Phi$. Mutual encircling of $(q_1,{\Phi }_1)$ and $(q_2,{\Phi }_2)$ produces the phase $\exp [i(q_1{\Phi }_2+q_2{\Phi }_1)/\hslash ]$ from the Aharonov-Bohm and Aharonov-Casher effects.

The Aharonov-Bohm effect is the phase $q\Phi /\hslash$ acquired when a charge $q$ encircles magnetic flux $\Phi$, even when the magnetic field vanishes along the charge's path.

The Aharonov-Casher effect is the electromagnetic dual in which a magnetic moment or flux acquires a phase by encircling electric charge.

Two particle types are mutual semions when taking either once around the other multiplies the state by $-1$.

An anyon condenses at a boundary when its string operator may terminate there without leaving an excitation, so the boundary can absorb or emit that anyon.

An electric boundary of the surface code condenses electric anyons, while a magnetic boundary condenses magnetic anyons. A local boundary Pauli can create a single condensed excitation because its string may end at that boundary.