Rigorous Analysis of Renormalization Group Pathologies in the 4-State   Clock Model
  We perform an exact renormalization-group analysis of one-dimensional 4-state clock models with complex interactions. Our aim is to provide a simple explicit illustration of the behavior of the renormalization-group flow in a system exhibiting a rich phase diagram. In particular we study the flow in the vicinity of phase transitions with a first-order character, a matter that has been controversial for years. We observe that the flow is continuous and single-valued, even on the phase transition surface, provided that the renormalized Hamiltonian exist. The characteristics of such a flow are in agreement with the Nienhuis-Nauenberg standard scenario, and in disagreement with the ``discontinuity scenario'' proposed by some authors and recently disproved by van Enter, Fern\'andez and Sokal for a large class of models (with real interactions). However, there are some points in the space of interactions for which a renormalized Hamiltonian cannot be defined. This pathological behavior is similar, and in some sense complementary, to the one pointed out by Griffiths, Pearce and Israel for Ising models. We explicitly see that if the transformation is truncated so as to preserve a Hamiltonian description, the resulting flow becomes discontinuous and multivalued at some of these points. This suggests a possible explanation for the numerical results that motivated the ``discontinuity scenario''. 
M. Asorey, J.G. Esteve, R. Fernandez J. Salas