[Paper Review] The Generic Critical Behaviour for 2D Polymer Collapse

Adam Nahum|arXiv (Cornell University)|Oct 30, 2015

Theoretical and Computational Physics3 citations

TL;DR

This paper resolves the long-standing debate on the universality of critical exponents at the 2D polymer $Θ$-point by demonstrating that the Duplantier-Saleur (DS) exponents are robust for non-self-crossing polymers via a $χ^{N-1}$ sigma model in the $N\to 1$ limit. It shows these exponents arise generically without fine-tuning, while self-crossing introduces an additional relevant perturbation, leading to different critical behavior governed by the generic tricritical $\mathrm{O}(n)$ model at $n=0$. The work also reveals a non-trivial operator degeneracy in the $\mathrm{CP}^{N-1}$ model for all $N$, with two operators of different symmetry having identical scaling dimensions.

ABSTRACT

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 $N ightarrow 1$ 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.

Motivation & Objective

• To resolve the longstanding debate on whether the Duplantier-Saleur (DS) critical exponents represent generic critical behavior at the 2D polymer $Θ$-point.
• To clarify the apparent paradox where two models in the same field theory class appear to have different numbers of relevant perturbations.
• To determine the stability of the DS fixed point under renormalization group flow in the presence and absence of polymer self-crossing.
• To analyze the operator content of the $\mathrm{CP}^{N-1}$ model at $N=1$, particularly the degeneracy of operators with distinct symmetries but identical scaling dimensions.

Proposed method

• Mapping the polymer collapse problem to the $\mathrm{CP}^{N-1}$ sigma model in the $N\to 1$ limit to analyze universality and stability.
• Using renormalization group analysis to count relevant perturbations and assess the robustness of the DS fixed point.
• Classifying symmetry-breaking perturbations in lattice models via their mapping to $\mathrm{SU}(N)$-symmetric field theories.
• Employing supersymmetry (SUSY) techniques to compute operator multiplicities and confirm the completeness of the operator spectrum at scaling dimension $x_4$.
• Analyzing the structure of irreducible and indecomposable representations in the SUSY formulation to explain the degeneracy of operators with different symmetry quantum numbers.
• Comparing the replica and SUSY formulations to confirm that scaling dimensions are preserved and that no additional operators exist at $x_4$.

Experimental results

Research questions

• RQ1 Are the Duplantier-Saleur critical exponents robust for generic non-self-crossing polymer models in two dimensions, or do they require fine-tuning?
• RQ2 Why do two models with identical field theory descriptions appear to have different numbers of relevant perturbations, suggesting contradictory stability conclusions?
• RQ3 How does the inclusion of polymer self-crossing alter the universality class and critical exponents of the $Θ$-point transition?
• RQ4 What explains the degeneracy of two operators with distinct symmetry properties but identical scaling dimensions in the $\mathrm{CP}^{N-1}$ model for all $N$?
• RQ5 Is there a marginal parity-odd operator in the $\mathrm{CP}^{N-1}$ model related to the winding angle, and how does it affect the critical behavior?

Key findings

• The Duplantier-Saleur critical exponents are robust and arise generically in non-self-crossing polymer models without fine-tuning, confirming their universality in two dimensions.
• The apparent paradox in perturbation counting arises because one of the models is fine-tuned due to an Ising-like order parameter, breaking the full $\mathrm{SU}(N)$ symmetry and invalidating naive universality assumptions.
• When polymer self-crossing is allowed, an additional independent relevant perturbation emerges, shifting the critical behavior to that of the generic tricritical $\mathrm{O}(n)$ model at $n=0$, distinct from the DS exponents.
• Two operators in the $\mathrm{CP}^{N-1}$ model—differing in symmetry—have identical scaling dimensions for all $N$, a result confirmed by matching multiplicities in the SUSY formulation.
• The $\mathrm{CP}^{N-1}$ model at $N=1$ hosts a marginal parity-odd operator related to the winding angle, which is not present in the replica formulation and may influence critical behavior.
• The complete set of operators at scaling dimension $x_4$ is identified and confirmed via SUSY representation theory, with total multiplicity matching the exact result from the replica approach.