Triangular Ising Antiferromagnet

Each square is a spin (black = up, white = down), with Hamiltonian \[ H = J \sum_{\langle i,j \rangle} s_i s_j - h \sum_i s_i, \qquad J = 1 \; (\text{antiferromagnetic}). \] The triangular lattice is drawn as a plain square grid with ordinary periodic boundary conditions, but each site couples to its N, S, E, W neighbors and its NW/SE diagonal neighbors — six bonds total, matching the connectivity of the triangular lattice. Each site together with its N and NW neighbors (or S and SE neighbors) forms an elementary triangle with three antiferromagnetic bonds, which can never all be satisfied at once — the model is geometrically frustrated. This macroscopic ground-state degeneracy prevents long-range order at any temperature, including \(T=0\) (Wannier 1950).

Updates are single-spin-flip Metropolis dynamics with respect to \(H\).