A new look at the collapse of two-dimensional polymers

TL;DR

This paper investigates the collapse behavior of two-dimensional polymers using an exactly solvable model, revealing complex conformal field theory structures and continuous critical exponents, and clarifies the nature of the theta point.

Contribution

It demonstrates that the continuum limit of the $ ext{Theta}_{ m BN}$ point involves non-compact degrees of freedom, providing new insights into the critical behavior of 2D polymer collapse.

Findings

The continuum limit of $ ext{Theta}_{ m BN}$} is a dense polymer with non-compact degrees of freedom.

Critical exponents vary continuously due to non-compact degrees of freedom.

Numerical checks support the theoretical predictions about the non-compact structure.

Abstract

We study the collapse of two-dimensional polymers, via an O($n$) model on the square lattice that allows for dilution, bending rigidity and short-range monomer attractions. This model contains two candidates for the theta point, $\Theta_{\rm BN}$ and $\Theta_{\rm DS}$, both exactly solvable. The relative stability of these points, and the question of which one describes the `generic' theta point, have been the source of a long-standing debate. Moreover, the analytically predicted exponents of $\Theta_{\rm BN}$ have never been convincingly observed in numerical simulations. In the present paper, we shed a new light on this confusing situation. We show in particular that the continuum limit of $\Theta_{\rm BN}$ is an unusual conformal field theory, made in fact of a simple dense polymer decorated with {\sl non-compact degrees of freedom}. This implies in particular that the critical exponents take continuous rather than discrete values, and that corrections to scaling lead to an unusual integral form. Furthermore, discrete states may emerge from the continuum, but the latter are only normalizable---and hence observable---for appropriate values of the model's parameters. We check these findings numerically. We also probe the non-compact degrees of freedom in various ways, and establish that they are related to fluctuations of the density of monomers. Finally, we construct a field theoretic model of the vicinity of $\Theta_{\rm BN}$ and examine the flow along the multicritical line between $\Theta_{\rm BN}$ and $\Theta_{\rm DS}$.