Universality class of the two-dimensional polymer collapse transition

Abstract

The nature of the θ 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 for an exactly solvable model. We use a representation of the problem via the CP N−1 σ model in the limit N → 1 to determine the stability of this critical point. First we prove that the Duplantier-Saleur (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. We also address 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-two-dimensional 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 from 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 (or, equivalently, any value of the loop fugacity). Also we argue that for any value of N the CP N−1 model has a marginal odd-parity operator that is related to the winding angle.

On truncations of the Chalker-Coddington model

“…Meanwhile, we see that, while the integrable model has different dimensions for operators inserting an odd number of loops, these operators in the unmodified truncated model seem to have the same dimensions as operators in the even sectors (cf Figure 5). Similar features have been encountered sometimes in the past [18,26].…”

Grand-canonical solution of semiflexible self-avoiding trails on the Bethe lattice

“…where the ratio R is given by Eq. (16). The bulk free energy per site on the BL, which differs from the one obtained before in Eq.…”

“…The inclusion of stiffness in the VISAW model, associating a bending energy to elementary bends of the chain, leads to an even richer phase diagram [15], with the tricritical points from both integrable models (DS and BN) residing on a multicritical line. The robustness of the DS exponents has also been discussed in a recent paper [16], where it has been shown that if crossings of the trails are included in the BN model, so that the lattice is no longer planar, the universality class is changed.…”

Two-dimensional O(n) models and logarithmic CFTs

“…Indeed, there are known examples of loop models with enhanced symmetry (often from O(n) to U(n)) where the loop crossing terms breaks the enhanced symmetry [75][76][77]. It was also shown in [41] that the low-T phase of the O(n) model for some integer n can be represented as a U(n)-symmetric model with an addition of some topological defects.…”