Universality class of the two-dimensional polymer collapse transition

Area of Science:

• Statistical Mechanics
• Polymer Physics
• Quantum Field Theory

Background:

• The nature of the theta point for 2D polymers and its critical exponents has been a long-standing debate.
• Previous work proposed candidates for critical exponents, including those by Duplantier and Saleur (DS) from an exactly solvable model.

Purpose of the Study:

• To determine the stability of the Duplantier-Saleur critical point for 2D polymers.
• To resolve theoretical questions regarding the robustness and universality of these exponents.
• To investigate the behavior of polymers with self-crossings and compare it to the non-crossing case.

Main Methods:

• Utilizing the CP^{N-1} sigma model in the N→1 limit to represent the polymer problem.
• Analyzing lattice models to prove the robustness of DS exponents for non-self-crossing polymers.
• Investigating the impact of self-crossings on critical exponents and comparing different theoretical models.

Main Results:

• The Duplantier-Saleur critical exponents are proven to be robust for non-self-crossing polymers, arising generically without fine-tuning.
• An apparent paradox concerning the stability of DS exponents in different lattice models is resolved by identifying a fine-tuned model.
• Self-crossing polymers exhibit different critical exponents, corresponding to the tricritical O(n) model at n=0, distinct from the non-crossing case.

Conclusions:

• The DS critical exponents are stable and broadly applicable to non-self-crossing 2D polymers.
• Self-crossing fundamentally alters the critical behavior, leading to a different universality class.
• The study also reveals insights into the operator content of the CP^{N-1} model, including operators with identical scaling dimensions and a marginal odd-parity operator related to winding angle.