Entropy crisis, ideal glass transition, and polymer melting: Exact solution on a Husimi cactus

We investigate an extension of the lattice model of melting of semiflexible polymers originally proposed by Flory. Along with a bending penalty epsilon, present in the original model and involving three sites of the lattice, we introduce an interaction energy epsilon(p), corresponding to the presence of a pair of parallel bonds and an interaction energy epsilon(h), associated with a hairpin turn. Both these new terms represent four-site interactions. The model is solved exactly on a Husimi cactus, which approximates a square lattice. We study the phase diagram of the system as a function of the energies. For a proper choice of the interaction energies, the model exhibits a first-order melting transition between a liquid and a crystalline phase at a temperature T-M. The continuation of the liquid phase below T-M gives rise to a supercooled liquid, which turns continuously into a new low-temperature phase, called metastable liquid, at T-MC