Decimations for one- and two-dimensional Ising and rotator models. II. Continuous vs discrete symmetries

We show how decimated Gibbs measures having unbroken continuous symmetry due to the Mermin-Wagner theorem, despite their discrete equivalents exhibiting phase transition, can still become non-Gibbsian. The mechanism rests on the occurrence of a spin-flop transition with a broken discrete symmetry, once the model is constrained by the decimated spins in a suitably chosen "bad"configuration.