On the universality of compact polymers

TL;DR

This paper investigates the critical behavior of fully packed loop models on various bipartite lattices, revealing new universality classes and establishing connections with known models like the O(n) and Potts models.

Contribution

It demonstrates that compact phases are limited to bipartite lattices and identifies the critical properties of the fully packed loop model on the square-octagon lattice.

Findings

Model is critical only for loop weights n < 1.88

Scaling limit matches the dense phase of the O(n) model

For n=2, equivalent to the selfdual 9-state Potts model

Abstract

Fully packed loop models on the square and the honeycomb lattice constitute new classes of critical behaviour, distinct from those of the low-temperature O(n) model. A simple symmetry argument suggests that such compact phases are only possible when the underlying lattice is bipartite. Motivated by the hope of identifying further compact universality classes we therefore study the fully packed loop model on the square-octagon lattice. Surprisingly, this model is only critical for loop weights n < 1.88, and its scaling limit coincides with the dense phase of the O(n) model. For n=2 it is exactly equivalent to the selfdual 9-state Potts model. These analytical predictions are confirmed by numerical transfer matrix results. Our conclusions extend to a large class of bipartite decorated lattices.