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We introduce an extension of the lattice model of melting of semiflexiblepolymers originally proposed by Flory. Along with a bending penalty, present inthe original model and involving three sites of the lattice, we introduce aninteraction energy that corresponds to the presence of a pair of parallel bondsand a second interaction energy associated with the presence of a hairpin turn.Both these new terms represent four-site interactions. The model is solvedexactly on a Husimi cactus, which approximates a square lattice. We study thephase diagram of the system as a function of the energies. For a proper choiceof the interaction energies, the model exhibits a first-order meltingtransition between a liquid and a crystalline phase. The continuation of theliquid phase below this temperature gives rise to a supercooled liquid, whichturns continuously into a new low-temperature phase, called metastable liquid.This liquid-liquid transition seems to have some features that arecharacteristic of the critical transition predicted by the mode-couplingtheory.Comment: To be published in Physical Review E, 68 (2) (2003

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