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.
Highlights
We are going to deal with the 2D O(3) model, a nonlinear σ model, which is known as the Heisenberg model, or CPð1Þ model
Topology can be defined in its lattice regularized version, but semiclassical arguments suggest that the topological susceptibility χt does not scale towards a finite continuum limit
There is a variety of models with topological sectors, and some of them are plagued by problems with the continuum scaling of χt
Summary
We are going to deal with the 2D O(3) model, a nonlinear σ model, which is known as the Heisenberg model, or CPð1Þ model. After a period of confusion, the consensus seemed to be that χtξ diverges at large ξ This conclusion is consistent with studies with different lattice actions in the same universality class [11,12,13]. The topological susceptibility can be assembled as χt 1⁄4 xhq0qxi, where solely the contact term at lattice site x 1⁄4 0 causes the divergence [10,13] This implies that it is an UVeffect, in agreement with the picture of more and more abundant tiny (with respect to ξ) topological windings as we approach the continuum limit [6]. When we increase the inverse coupling β, which defines the standard continuum limit, say at fixed physical size L=ξ If this leads to a divergence of χtξ, there is no way to renormalize it by subtracting counterterms. GF times) have been published in two proceedings contributions [21]
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