Collapse of a polymer in two dimensions

TL;DR

This paper uses transfer-matrix calculations to study how self-attraction affects the critical behavior of two-dimensional polymers, revealing a crossover from self-avoiding walk to theta point behavior.

Contribution

It introduces a numerical transfer-matrix approach based on the O(n) loop model to analyze phase transitions in 2D polymers with self-attraction.

Findings

Identifies phase diagram as a function of chemical potential and attraction strength.

Observes crossover from self-avoiding walk to theta point universality class.

Provides finite-size scaling analysis of magnetic correlation length.

Abstract

We numerically investigate the influence of self-attraction on the critical behaviour of a polymer in two dimensions, by means of an analysis of finite-size results of transfer-matrix calculations. The transfer matrix is constructed on the basis of the O($n$) loop model in the limit $n \to 0$. It yields finite-size results for the magnetic correlation length of systems with a cylindrical geometry. A comparison with the predictions of finite-size scaling enables us to obtain information about the phase diagram as a function of the chemical potential of the loop segments and the strength of the attractive potential. Results for the magnetic scaling dimension can be interpreted in terms of known universality classes. In particular, when the attractive potential is increased, we observe the crossover between polymer critical behaviour of the self-avoiding walk type to behaviour described earlier for the theta point.