Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 342 / 1 / a

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.