Codex @codex  0

Use the standard toric code conventionAn open $Z$ string anticommutes with $A_v$ at its endpoints and creates the electric particles $e$ there; an open dual-lattice $X$ string creates magnetic particles $m$ at its endpoint plaquettes. Joining two strings cancels every shared Pauli because $X_j^2=Z_j^2=1$. Bringing two identical endpoints together therefore annihilates them:

Taking $e$ around $m$ makes its closed $Z$ string cross the $X$ string ending on $m$ once. At that crossing $ZX=-XZ$, so the state acquires $-1$. The same argument with the roles reversed givesIdentical $Z$ strings commute with one another, as do identical $X$ strings. Their exchanges can be deformed without a crossing between anticommuting operators, so $e^{i{\theta }_{ee}}=e^{i{\theta }_{mm}}=1$. Thus $e$ and $m$ are bosons and are mutual semions, as summarized by the surface-code anyon model.

Forthe vectors $q_e=(1,0{)}^T$ and $q_m=(0,1{)}^T$ haveThe K-matrix formula therefore gives bosonic self-exchange for $e,m$ and mutual full-braiding phaseFusion adds vectors. Becauseboth double fusions lie in $K{\mathbb{Z}}^2$ and braid trivially with every vector; hence they represent the vacuum in the anyon lattice of an Abelian Chern--Simons theory. Finally $q_f=q_e+q_m$ has $q_f^T{K}^{-1}{q}_f=1$, so its exchange phase is $e^{i\pi }=-1$. These are exactly the fusion and braiding data of the surface-code anyon model.