Perfect topological charge for asymptotically free theories

TL;DR

This paper demonstrates that in asymptotically free lattice models, a perfect topological charge can be defined without defects, and shows that topological susceptibility may not be a meaningful physical quantity in these models.

Contribution

It introduces a scale-invariant, defect-free topological charge for asymptotically free lattice theories and applies it to measure topological susceptibility in the $d=2$ O(3) model.

Findings

Topological charge can be defined without defects in these models.

Topological susceptibility appears to be non-physical in the studied model.

High-precision measurements across various correlation lengths.

Abstract

The classical equations of motion of the perfect lattice action in asymptotically free $d=2$ spin and $d=4$ gauge models possess scale invariant instanton solutions. This property allows the definition of a topological charge on the lattice which is perfect in the sense that no topological defects exist. The basic construction is illustrated in the $d=2$ O(3) non--linear $\sigma$--model and the topological susceptibility is measured to high precision in the range of correlation lengths $\xi \in (2 - 60)$. Our results strongly suggest that the topological susceptibility is not a physical quantity in this model.