Is it right, that non-integrable quantum spin models in one dimension become integrable at their critical points? Or do they stay nonintegrable at the critical point also? Are there any examples known? In the field of 2d classical models, the three-state Potts model is not in general integrable, but this model is integrable at the critical point.
If the non-integrable quantum spin chain at the critical points can be described as a conformal field theory (not always the case), we can say that the model is "integrable''. Because CFT can be seen as an "integrable'' theory since it can be solved exactly and Yang-Baxter relation is satisfied naturally. If the critical points cannot be described by CFT, there is no general guarantee whether the underlying field theory is integrable or not.