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 two basic particles have bosonic self-statistics and full mutual-braid phase $-1$. Moreover $|\det K_1|=4$. These are the electric and magnetic particles of the surface code.

Forthe first basic particle is a fermion, the second is a boson, and they are mutual semions; again $|\det K_2|=4$. They can be identified with $f=e\times m$ and $m$, so this is merely another integral basis for the same surface-code phase.

Finally,which already rules it out. Therefore

An $r_x$ string is a product of X operators, while a $g_z$ string is a product of Z operators. A process that carries $r_x$ once around $g_z$ can be represented by a closed red X string crossing the green Z string once. At the crossing qubit,while all other factors commute. Reversing the order of the two string operators therefore multiplies the state by $-1$. The full braid phase isso $r_x$ and $g_z$ are mutual semions.

Violating a star stabilizer $A_v$ creates an electric $e$ particle; violating a plaquette stabilizer $B_p$ creates a magnetic $m$ particle. Open Z strings create pairs of $e$ endpoints, while open X strings create pairs of $m$ endpoints. Their fusion $ϵ=e\times m$ is the third nontrivial particle type.

Two equal-type strings can be exchanged without a phase, so $e$ and $m$ have bosonic self-statistics. An X string and a Z string crossing once anticommute, so taking $e$ around $m$ gives phase $-1$: they are mutual semions. Combining these two mutual-semion species givesThus $ϵ$ is a fermion.