Nematic Defect Annihilation — ±½ Topological Defects

Liquid crystal defects: In a nematic, the director n̂ has head-tail symmetry (n̂≡-n̂), so the fundamental group is π₁(RP²)=Z₂, allowing half-integer topological defects. A +½ defect (comet-shaped) and a -½ defect (trefoil) attract and annihilate, releasing energy ∝ln(r/a). Active nematics break Onsager symmetry: +½ defects self-propel (they have a polarity); -½ defects are stationary on average. Coarsening law: ⟨N⟩ ∝ t⁻¹ (2D defect-antidefect annihilation).