Theta-point polymers in the plane and Schramm-Loewner evolution

Area of Science:

• Statistical mechanics
• Polymer physics
• Conformal field theory

Background:

• Polymers at the theta temperature exhibit universal behavior.
• Schramm-Loewner evolution (SLE) models critical curves in 2D statistical physics.
• Tricritical polymers represent a specific phase transition point.

Purpose of the Study:

• To establish a connection between lattice polymer models at the theta temperature and Schramm-Loewner chains in the continuum.
• To develop an exact sampling algorithm for tricritical polymers, excluding finite-chain effects.
• To verify theoretical predictions for critical exponents and polymer properties.

Main Methods:

• Utilizing a radial implementation of the zipper method to compute the driving function from the lattice model.
• Comparing lattice polymer properties with those derived from Schramm-Loewner chains.
• Calculating correlation length exponent (ν) and correction-to-scaling exponent (Δ₁).

Main Results:

• The driving function from the lattice model converges to Brownian motion with diffusivity κ=6.
• The distribution function of internal polymer portions is well approximated by SLE results.
• Measured exponents ν=4/7 and Δ₁=72/91 align with theoretical predictions for the theta point and percolation, respectively.
• Shape factor and asphericity show accord with theta-point end-to-end values.

Conclusions:

• The study establishes a strong link between lattice polymers at the theta temperature and Schramm-Loewner chains.
• The developed method provides an effective tool for studying tricritical polymers.
• The findings support theoretical predictions for critical phenomena in polymer systems.