Meron-Cluster Simulation of the $\theta$-Vacuum in the 2-d $O(3)$-Model

The 2-d $O(3)$-model with a $\theta$-vacuum term is formulated in terms of Wolff clusters. Each cluster carries a half-integer topological charge. The clusters with charge $\pm 1\mathrm{/}2$ are identified as merons. At $\theta =\pi$ the merons are bound in pairs inducing a second order phase transition at which the mass-gap vanishes. The construction of an improved estimator for the topological charge distribution makes numerical simulations of the phase transition feasible. The measured critical exponents agree with those of the $k = 1$ Wess-Zumino-Novikov-Witten model. Our results are consistent with Haldane’s conjecture for 1-d antiferromagnetic quantum spin chains.