Interpolating between O(N)-symmetric σ-models with N = 1, 2, 3

We interpolate between the O(N)- and the O(N-1)-symmetric non-linear σ-models with standard action by suppressing the Nth component of the n-vector field with a mass term. We find a critical line in the coupling-mass plane connecting the critical point of the Ising model (N=1) with the critical point of the XY-model (N=2). This line extends towards the region of non-monotonous behaviour of the β-function of the O(3)-symmetric model. The massless phase of the XY-model is separated from the ordered phase of the Ising model by a transition at zero mass, while it is separated from the only phase of the O(3)-model by a transition that seems to occur at exponentially small mass. (orig.)