Non-trivial θ -vacuum effects in the 2-d O (3) model

We study θ -vacua in the 2-d lattice O (3) model using the standard action and an optimized constraint action with very small cut-off effects, combined with the geometric topological charge. Remarkably, dislocation lattice artifacts do not spoil the non-trivial continuum limit at θ 0 , and there are different continuum theories for each value 0 ≤ θ ≤ π . A very precise Monte Carlo study of the step scaling function indirectly confirms the exact S-matrix of the 2-d O (3) model at θ = π.