The Generic Critical Behaviour for 2D Polymer Collapse

The nature of the theta point for a polymer in two dimensions has long been debated, with a variety of candidates put forward for the critical exponents. This includes those derived by Duplantier and Saleur (DS) for an exactly solvable model. We use a representation of the problem via the $CP^{N-1}$ sigma model in the limit $N \rightarrow 1$ to determine the stability of this critical point. First we prove that the DS critical exponents are robust, so long as the polymer does not cross itself: they can arise in a generic lattice model, and do not require fine tuning. This resolves a longstanding theoretical question. However there is an apparent paradox: two different lattice models, apparently both in the DS universality class, show different numbers of relevant perturbations, apparently leading to contradictory conclusions about the stability of the DS exponents. We explain this in terms of subtle differences between the two models, one of which is fine-tuned (and not strictly in the DS universality class). Next, we allow the polymer to cross itself, as appropriate e.g. to the quasi-2D case. This introduces an additional independent relevant perturbation, so we do not expect the DS exponents to apply. The exponents in the case with crossings will be those of the generic tricritical $O(n)$ model at $n=0$, and different to the case without crossings. We also discuss interesting features of the operator content of the $CP^{N-1}$ model. Simple geometrical arguments show that two operators in this field theory, with very different symmetry properties, have the same scaling dimension for any value of $N$ (equivalently, any value of the loop fugacity). Also we argue that for any value of $N$ the $CP^{N-1}$ model has a marginal parity-odd operator which is related to the loops' winding angle.