Z₂ Gauge Theory — Toric Code Fractionalization
Plaquette (B_p) & Vertex (A_v) Operators · e and m Anyons · String Operators
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Grid size L:
e-anyon pairs:
1
m-anyon pairs:
1
Show strings:
on
Create e-anyon pair
Create m-anyon pair
Braid e around m
Reset ground state
Energy
Toric Code:
H=−Σ_v A_v − Σ_p B_p where A_v=∏σ^x, B_p=∏σ^z on vertex/plaquette qubits.
e-anyons:
Vertex violations (A_v=−1), created by Z-string operators.
m-anyons:
Plaquette violations (B_p=−1), created by X-string operators.
Statistics:
Braiding e around m gives phase π (mutual semionic statistics), implementing Z₂ fractionalization.