Entropy Crisis, Ideal Glass Transition and Polymer Melting: Exact Solution on a Husimi Cactus

We introduce an extension of the lattice model of melting of semiflexible polymers originally proposed by Flory. Along with a bending penalty, present in the original model and involving three sites of the lattice, we introduce an interaction energy that corresponds to the presence of a pair of parallel bonds and a second interaction energy associated with the presence of 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. The continuation of the liquid phase below this temperature gives rise to a supercooled liquid, which turns continuously into a new low-temperature phase, called metastable liquid. This liquid-liquid transition seems to have some features that are characteristic of the critical transition predicted by the mode-coupling theory.