Continuously Varying Exponents for Oriented Self-Avoiding Walks

TL;DR

This paper uses conformal field theory to show that in oriented self-avoiding walks, the number of configurations varies continuously with energy differences, while the fractal dimension remains fixed.

Contribution

It introduces a conformal field theory framework demonstrating continuous variation of the exponent b3 in oriented polymers due to a specific perturbation.

Findings

The exponent b3 varies continuously with energy differences.

The fractal dimension bd remains fixed despite the variation.

A line of fixed points is identified in the theory.

Abstract

A two-dimensional conformal field theory with a conserved $U(1)$ current $\vec J$, when perturbed by the operator ${\vec J}^{\,2}$, exhibits a line of fixed points along which the scaling dimensions of the operators with non-zero $U(1)$ charge vary continuously. This result is applied to the problem of oriented polymers (self-avoiding walks) in which the short-range repulsive interactions between two segments depend on their relative orientation. While the exponent $\nu$ describing the fractal dimension of such walks remains fixed, the exponent $\gamma$, which gives the total number $\sim N^{\gamma-1}\mu^N$ of such walks, is predicted to vary continuously with the energy difference.