Hyperscaling for polymer rings

TL;DR

This paper investigates the hyperscaling properties of polymer rings on lattices, calculating universal amplitudes using the Edwards model, and confirms results through field theory and simulations, providing new insights into critical phenomena.

Contribution

It introduces a calculation of universal amplitudes for polymer rings using the Edwards model and verifies these findings with field theory and simulations.

Findings

Universal amplitude A(d) derived from Edwards model

Confirmation of hyperscaling relations for polymer rings

Explicit expression for the universal constant λ

Abstract

The statistics of a long closed self-avoiding walk (SAW) or polymer ring on a $ d $-dimensional lattice obeys hyperscaling. The combination $ p_N \left\langle R^2 \right\rangle^{ d/2}_N\mu^{ -N}, $ (where $ p_N $ is the number of configurations of an oriented and rooted $ N $-step ring, $ \left\langle R^2 \right\rangle_ N $ a typical average size squared, and $ \mu $ the SAW effective connectivity constant of the lattice) is equal for $ N \longrightarrow \infty $ to a lattice-dependent constant times a universal amplitude $ A(d). $ The latter amplitude is calculated directly from the minimal continuous Edwards model to second order in $ \varepsilon \equiv 4-d. $ The case of rings at the upper critical dimension $ d=4 $ is also studied. The results are checked against field theoretical calculations, and former simulations. As a consequence, we show that the universal constant $ \lambda $ appearing to second order in $ \varepsilon $ in all critical phenomena amplitude ratios is equal to $ \lambda = {1 \over 18} \left[\psi^{ \prime}( 1/6)+\psi^{ \prime}( 1/3) \right]-{4\pi^ 2 \over 27}. $