Universal Properties of Self-Avoiding Walks from Two-Dimensional Field Theory

TL;DR

This paper uses a 2D field theory approach to derive universal properties of self-avoiding walks, providing theoretical predictions that align with lattice studies and revealing that low-particle approximations are highly accurate.

Contribution

It presents the first theoretical prediction of the amplitude ratio C/D for self-avoiding chains and loops, supported by field theoretic derivations and comparison with numerical data.

Findings

Predicted amplitude ratio C/D matches lattice results within errors.

Low-m particle state sums (m≤2) yield highly accurate results.

Softening of branch cuts explains the effectiveness of low-particle approximations.

Abstract

We use the recently conjectured exact $S$-matrix of the massive ${\rm O}(n)$ model to derive its form factors and ground state energy. This information is then used in the limit $n\to0$ to obtain quantitative results for various universal properties of self-avoiding chains and loops. In particular, we give the first theoretical prediction of the amplitude ratio $C/D$ which relates the mean square end-to-end distance of chains to the mean square radius of gyration of closed loops. This agrees with the results from lattice enumeration studies to within their errors, and gives strong support for the various assumptions which enter into the field theoretic derivation. In addition, we obtain results for the scaling function of the structure factor of long loops, and for various amplitude ratios measuring the shape of self-avoiding chains. These quantities are all related to moments of correlation functions which are evaluated as a sum over $m$-particle intermediate states in the corresponding field theory. We show that in almost all cases, the restriction to $m\leq2$ gives results which are accurate to at least one part in $10^3$. This remarkable fact is traced to a softening of the $m>2$ branch cuts relative to their behaviour based on phase space arguments alone, a result which follows from the threshold behaviour of the two-body $S$-matrix, $S(0)=-1$. Since this is a general property of interacting 2d field theories, it suggests that similar approximations may well hold for other models. However, we also study the moments of the area of self-avoiding loops,