Ising anyons are non-Abelian quasiparticles: braiding two of them applies a unitary matrix (not just a phase) to the degenerate ground state. This is topological quantum computation.
The Ising model has three anyon types: 1 (vacuum), σ (Ising anyon), ψ (fermion). Fusion rules: σ×σ = 1+ψ, σ×ψ = σ.
Braiding σ around σ gives the matrix: R = e^{iπ/8} · (1/√2) [[1, i],[i, 1]] — a √NOT gate. Two braidings = NOT. Topologically protected from local errors.