Geometric Frustration — Triangular Ising

Antiferromagnet on triangular lattice: ground state entropy S ~ N·k·ln(4/3)^½ (Wannier 1950)

Energy: — | Frustrated bonds: — | Magnetization: —
On the triangular lattice, antiferromagnetic bonds (J < 0) cannot all be satisfied simultaneously — at least one frustrated bond per triangle is unavoidable. This geometric frustration leads to extensive ground state degeneracy: S₀ = Nk·ln(W)^(1/N) ≈ Nk·0.323 (Wannier 1950), explaining why the triangular AFM has no phase transition. Compare with the square lattice (right), where AFM order is unfrustrated and a Néel state exists. Red bonds = frustrated.