Topological Susceptibility of the 2d O(3) Model under Gradient Flow

TL;DR

This study examines whether the topological susceptibility in the 2d O(3) model remains divergent or scales to a finite value when configurations are smoothed using Gradient Flow, finding no evidence of continuum scaling.

Contribution

It provides the first detailed analysis of the effect of Gradient Flow on topological susceptibility scaling in the 2d O(3) model.

Findings

Gradient Flow reduces topological susceptibility on fine lattices.

Even with extensive smoothing, susceptibility does not show continuum scaling.

Divergence of susceptibility persists despite smoothing efforts.

Abstract

The 2d O(3) model is widely used as a toy model for ferromagnetism and for Quantum Chromodynamics. With the latter it shares --- among other basic aspects --- the property that the continuum functional integral splits into topological sectors. Topology can also be defined in its lattice regularised version, but semi-classical arguments suggest that the topological susceptibility $\chi_{\rm t}$ does not scale towards a finite continuum limit. Previous numerical studies confirmed that the quantity $\chi_{\rm t}\, \xi^{2}$ diverges at large correlation length $\xi$. Here we investigate the question whether or not this divergence persists when the configurations are smoothened by the Gradient Flow (GF). The GF destroys part of the topological windings; on fine lattices this strongly reduces $\chi_{\rm t}$. However, even when the flow time is so long that the GF impact range --- or smoothing radius --- attains $\xi/2$, we do still not observe evidence of continuum scaling.