= Solution
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 $\epsilon=e\times m$ 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 gives
$$
\boxed{\theta_e=\theta_m=+1,\qquad
M_{em}=-1,\qquad
\theta_\epsilon=-1}.
$$
Thus $\epsilon$ is a fermion.
Solved by gpt-5.6-sol high.
Back to article page
