An improved perturbation approach to the 2D Edwards polymer -- corrections to scaling

TL;DR

This paper introduces a new perturbation method for 2D Edwards polymers that accurately predicts the long-chain behavior and determines the correction-to-scaling exponent as 1/2, improving upon previous approaches.

Contribution

It presents a novel perturbation approach starting from the ground state, enabling precise calculation of the correction-to-scaling exponent in 2D polymer models.

Findings

Convergent calculation of mean square end-to-end distance for large N.

Determination of the correction-to-scaling exponent Δ as 1/2.

Support from analysis of 2D self-avoiding walks on the continuum.

Abstract

We present the results of a new perturbation calculation in polymer statistics which starts from a ground state that already correctly predicts the long chain length behaviour of the mean square end--to--end distance $\langle R_N^2 \rangle\ $, namely the solution to the 2~dimensional~(2D) Edwards model. The $\langle R_N^2 \rangle$ thus calculated is shown to be convergent in $N$, the number of steps in the chain, in contrast to previous methods which start from the free random walk solution. This allows us to calculate a new value for the leading correction--to--scaling exponent~$\Delta$. Writing $\langle R_N^2 \rangle = AN^{2\nu}(1+BN^{-\Delta} + CN^{-1}+...)$, where $\nu = 3/4$ in 2D, our result shows that $\Delta = 1/2$. This value is also supported by an analysis of 2D self--avoiding walks on the {\em continuum}.