N=2 Supersymmetry, Painleve III and Exact Scaling Functions in 2D Polymers

TL;DR

This paper explores the off-critical 2D $O(n)$ model, revealing connections between supersymmetry, Painleve III equations, and polymer scaling functions, supported by numerical simulations and extending known ground-state energy results.

Contribution

It demonstrates the relation of polymer scaling functions to Painleve III equations using N=2 supersymmetry and extends ground-state energy results to non-unitary minimal models.

Findings

Scaling function for one polymer loop relates to Painleve III solution.

Ground-state energy results confirmed numerically.

Effective central charge exhibits non-monotonous flow with roaming behavior.

Abstract

We discuss in this paper various aspects of the off-critical $O(n)$ model in two dimensions. We find the ground-state energy conjectured by Zamolodchikov for the unitary minimal models, and extend the result to some non-unitary minimal cases. We apply our results to the discussion of scaling functions for polymers on a cylinder. We show, using the underlying N=2 supersymmetry, that the scaling function for one non-contractible polymer loop around the cylinder is simply related to the solution of the Painleve III differential equation. We also find the ground-state energy for a single polymer on the cylinder. We check these results by numerically simulating the polymer system. We also analyze numerically the flow to the dense polymer phase. We find there surprising results, with a $c_{\hbox{eff}}$ function that is not monotonous and seems to have a roaming behavior, getting very close to the values 81/70 and 7/10 between its UV and IR values of 1.