Abelian Anyons

The simplest anyons act on the quantum state by multiplying it by a phase. Because phases commute, these are called abelian anyons. The toric code realizes the canonical example, and analyzing it shows how braiding produces a topologically invariant phase — a discrete analogue of the Aharonov–Bohm effect.

Two species in the toric code

The toric code has two basic nontrivial excitations: electric charges $e$ (violations of the vertex/star stabilizers $A_v=\prod_{i\in v} X_i$ ) and magnetic fluxes $m$ (violations of the plaquette stabilizers $B_p=\prod_{i\in p} Z_i$ ). Each is created in pairs at the ends of an error string: an $e$ pair at the ends of a $Z$ -string, an $m$ pair at the ends of an $X$ -string. Both $e$ and $m$ are their own antiparticles ( $e \times e = 1$ , $m \times m = 1$ ), and each individually behaves as a boson.

Mutual statistics: the $-1$ of braiding

The interesting statistics are mutual: between an $e$ and an $m$ . Move an $m$ flux on a closed loop that encircles a single $e$ charge. The $e$ excitation is created by a $Z$ -string and the $m$ -transport is an $X$ -string; the loop crosses the $e$ 's string an odd number of times, and the anticommutation $X Z = -Z X$ at the single crossing site contributes

\langle \psi | \,(\text{braid})\, | \psi \rangle = -\,\langle \psi | \psi \rangle.

So carrying an $m$ all the way around an $e$ multiplies the state by $-1$ :

|\psi\rangle \;\longmapsto\; -|\psi\rangle.

The bound state $\varepsilon = e \times m$ inherits these crossings and turns out to be a fermion: a self-exchange of two $\varepsilon$ 's gives a $-1$ . The toric code thus hosts a $\mathbb{Z}_2 \times \mathbb{Z}_2$ fusion structure with four superselection sectors $\{1, e, m, \varepsilon\}$ , often called a $\mathbb{Z}_2$ gauge theory or $D(\mathbb{Z}_2)$ anyon model.

A topological Aharonov–Bohm phase

The full braid (one particle taken once around the other) gives the mutual statistics phase $e^{i\theta_{em}}$ with $\theta_{em}=\pi$ for the toric code. The decisive feature is that this phase depends only on the winding number — how many times the loop encircled the other anyon — and not on the loop's size or shape. It is the topological analogue of the Aharonov–Bohm phase a charge picks up encircling a flux tube: only the enclosed flux matters, not the trajectory. Here the "charge" and "flux" are emergent quasiparticles of the medium, and the enclosed "flux" is quantized to give exactly $-1$ .

Why a single state, not a degenerate space

Because every braid is a phase, a configuration of abelian anyons at fixed positions has a unique state up to that phase. There is no protected degenerate space among the quasiparticles themselves. Storage of quantum information instead uses the global ground-state degeneracy on a nontrivial surface (the $4^g$ states on a genus- $g$ surface) — this is the logical space of the surface/toric code. Abelian braiding alone is therefore not enough for universal computation, motivating the non-abelian anyons of the next lessons.

What to take away

Abelian anyons braid by multiplying the state by a phase that depends only on winding number. The toric code's $e$ and $m$ are each bosons but braid with mutual statistics $-1$ : carrying one fully around the other gives a phase of $-1$ , a topological Aharonov–Bohm effect enforced by $XZ=-ZX$ . Information is stored not in the anyons but in the global ground-state degeneracy of the surface.