Exact Results for the Adsorption of a Flexible Self-Avoiding Polymer Chain in Two Dimensions

TL;DR

This paper provides exact critical couplings and universal exponents for the adsorption transition of a flexible self-avoiding polymer in two dimensions, using Bethe Ansatz solutions of the O(n) loop model.

Contribution

It derives exact critical parameters and confirms conjectures about geometric scaling dimensions for polymer adsorption on the honeycomb lattice.

Findings

Exact critical couplings for adsorption transition derived.

Universal critical exponents confirmed through Bethe Ansatz.

Geometric scaling dimensions match conjectured values in Kac formula.

Abstract

We derive the exact critical couplings ($x^*, y_{\rm a}^*$), where $y_{\rm a}^*/x^* = \sqrt{1+\sqrt2} = 1.533\ldots\,$, for the polymer adsorption transition on the honeycomb lattice, along with the universal critical exponents, from the Bethe Ansatz solution of the O($n$) loop model at the special transition. Our result for the thermal scaling dimension, and thus the crossover exponent $\phi=\frac{1}{2}$, is in agreement with an earlier result based on conformal invariance arguments. Our result for the geometric scaling dimensions confirms recent conjectures that they are given by $h_{\ell+1,3}$ in the Kac formula.