Singularrenormalization group transformations and first order phase transitions (I)

It is argued that standard position and momentum space renormalization group (RG) transformations are singular (i.e. lead to a singular hamiltonian after a finite number of RG steps) in large regions of the coupling constant space. It is shown in the d = 3, φ⁶ O(N)_N→∞ model that the momentum space RG transformation is singular in all those points of the coupling constant space, where metastable states exist. This region includes the full first order phase transition surface and its neighbourhood. Several other examples are discussed to illustrate that this phenomenon is generic and not a specific large N effect. Some earlier and recent anomalous Monte Carlo renormalization group results are consistent with this conclusion.